BiologyISC Class 11 · Plant Growth and Development

Plant Growth: Phases, Sigmoid Curve, Growth Rates and Differentiation

Updated 17 October 2026 · Fact-checked

Plant growth is an irreversible increase in size, mostly indeterminate, that takes place in meristematic, elongation and maturation phases. It follows a sigmoid curve (lag, log, stationary). Arithmetic growth is Lt = L₀ + rt; geometric growth is W₁ = W₀eʳᵗ. Development covers growth, differentiation, dedifferentiation and redifferentiation.

Notice the curve is slow in the lag phase, steepest in the log phase, then flattens in the stationary phase as nutrients run short.

Illustration drawn for QuestionBuddy.

Understand Growth Phases, Growth Rates and Measurement

Growth. An irreversible, permanent increase in size, volume or dry weight, brought about by cell division, enlargement and differentiation, needing water, oxygen and nutrients (and a suitable temperature). Plant growth is open: it continues throughout life from meristems (indeterminate growth), unlike the limited growth of most animals.

Phases. *Meristematic phase:* cell division at the root and shoot tips. *Elongation phase:* cells enlarge by vacuolation and new wall deposition (the region just behind the tip). *Maturation phase:* cells attain their final size and are differentiated.

Growth curve. Over time growth is sigmoid: a lag phase (slow), a log or exponential phase (rapid), and a stationary phase (limited nutrients slow growth).

Rates. *Arithmetic growth:* only one daughter cell keeps dividing; length increases linearly, Lₜ = L₀ + rt, where r is the growth rate (elongation per unit time), e.g. a root elongating at a constant rate. *Geometric growth:* both daughter cells divide; W₁ = W₀eʳᵗ (e = base of natural logarithms, r = relative growth rate). *Absolute growth rate:* total growth per unit time; *relative growth rate:* growth per unit time per unit initial parameter.

Measurement. Direct measurement with a ruler; indirectly with an auxanometer, in which the plant tip is attached by a thread over a pulley to a pointer moving on an arc scale (magnification equals the ratio of the pointer arm to the pulley radius).

Differentiation and development. Differentiation: cells derived from meristems mature into different types. Dedifferentiation: living differentiated cells regain the capacity to divide (interfascicular cambium and cork cambium from parenchyma). Redifferentiation: dedifferentiated cells lose division ability and mature again. Development = growth + differentiation; plants show plasticity (heterophylly in cotton and coriander). Seed germination may be epigeal (cotyledons above ground: castor, bean) or hypogeal (cotyledons remain in the soil: maize, pea, gram); vivipary is germination within the fruit (mangroves).

Growth Phases, Growth Rates and Measurement: key facts

Arithmetic growth: $L_{t} = L_{0} + r t$; r = growth per unit time (linear growth).
Geometric growth: $W_{1} = W_{0} e^{(} r t)$; r = relative growth rate; e = 2.718.
Relative growth rate: $r = l n (W_{1} / W_{0}) / t$; From the geometric growth equation.
Auxanometer: $G r o w t h = p o in t er m o v e m e n t / ma g ni f i c a t i o n$; Magnification = pointer arm/pulley radius.

Growth Phases, Growth Rates and Measurement in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

A shoot measured over a week

Suppose a bean seedling weighs 2.0 g on day 0 and 5.0 g after 10 days of exponential growth.

Find its relative growth rate per day (ln 2.5 = 0.916).

Try it first, then show the solutionSolution

W₁ = W₀eʳᵗ gives r = ln(W₁/W₀)/t.
r = ln(5.0/2.0)/10 = ln 2.5/10.
r = 0.916/10 = 0.092 per day (9.2% per day).

Answer: About 0.092 per day.

Memory hook: Geometric growth uses the exponential equation.

Tricks to remember Growth Phases, Growth Rates and Measurement

Short memory aids. Say each one aloud once, then use it on a question the same day.

Linear is arithmetic, exponential is geometric
One cell keeps dividing: arithmetic. Both cells divide: geometric.
Use it when: Choosing the growth equation.
Epi up, hypo stay
Epigeal cotyledons rise above the soil; hypogeal ones stay below.
Use it when: Germination types.

Growth Phases, Growth Rates and Measurement: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Type of growth in plants · 1 mark · MCQBoard level

The type of growth shown by plants due to the activity of meristems throughout life is called:
(a) determinate
(b) indeterminate (open)
(c) arithmetic
(d) restricted

Try it first, then show the solutionSolution

Plants continue to grow from meristems, hence open or indeterminate growth.

Answer: (b) indeterminate (open)

Example 2: Phases of growth · 3 marks · Short answerBoard level

Name the three phases of growth and describe the changes in each. What is the sigmoid growth curve?

Try it first, then show the solutionSolution

Meristematic phase: cell division at the tips. (0.75 mark)
Elongation phase: enlargement of cells by vacuolation and new wall material. (0.75 mark)
Maturation phase: cells reach final size and differentiate. (0.75 mark)
Sigmoid curve: lag, log and stationary phases of growth against time. (0.75 mark)

Answer: Meristematic, elongation, maturation; the S-shaped growth curve.

Example 3: Arithmetic growth · 3 marks · Short answerHarder board level

A root grows from 2.0 cm to 14.0 cm in 6 days at a constant rate. Find r and the length after 10 days. Is the growth arithmetic or geometric?

Try it first, then show the solutionSolution

r = (14.0 − 2.0)/6 = 2.0 cm per day. (1 mark)
L₁₀ = L₀ + rt = 2.0 + 2.0 × 10 = 22 cm. (1 mark)
Constant increment per unit time: arithmetic growth. (1 mark)

Answer: r = 2 cm per day; 22 cm; arithmetic.

Example 4: Geometric growth · 4 marks · Short answerHarder board level

A culture weighs 2.0 g and 5.0 g after 10 days of exponential growth. Find the relative growth rate (ln 2.5 = 0.916) and the expected weight after 20 days (e^1.832 = 6.25).

Try it first, then show the solutionSolution

r = ln(W₁/W₀)/t = 0.916/10 = 0.0916 per day. (1.5 marks)
W₂₀ = W₀e^(rt) = 2.0 × e^(0.0916 × 20) = 2.0 × e^1.832. (1.5 marks)
= 2.0 × 6.25 = 12.5 g (the weight grows 2.5 times each 10 days: 2 → 5 → 12.5). (1 mark)

Answer: r = 0.092 per day; 12.5 g after 20 days.

Example 5: Differentiation and germination · 5 marks · Long answerHOTS / competency

Explain differentiation, dedifferentiation and redifferentiation with one example each. Differentiate epigeal and hypogeal germination with an example each. Why is vivipary in mangroves an advantage?

Try it first, then show the solutionSolution

Differentiation: cells from meristems mature into specific types (xylem vessels). (1 mark)
Dedifferentiation: living differentiated cells regain division ability (cork cambium and interfascicular cambium from parenchyma). (1 mark)
Redifferentiation: cells produced by dedifferentiation lose the capacity to divide and mature again (secondary xylem). (1 mark)
Epigeal: cotyledons come above the ground (bean, castor); hypogeal: cotyledons stay underground (maize, pea). (1 mark)
Vivipary: seeds germinate on the plant, so the seedling falls into the mud and establishes quickly in unstable, saline soil. (1 mark)

Answer: Differentiation, dedifferentiation, redifferentiation; epigeal and hypogeal; vivipary suits marshy soil.

Where students lose marks in Growth Phases, Growth Rates and Measurement

The slips examiners see most often, and what to write instead.

Mistake: Confusing arithmetic and geometric growth
Using the exponential equation for linear growth.
Do this: Linear growth: Lₜ = L₀ + rt; exponential: W₁ = W₀eʳᵗ.
Usually costs: 1 to 2 marks.
Mistake: Mixing epigeal and hypogeal
Giving maize as epigeal.
Do this: Maize and pea are hypogeal; bean and castor are epigeal.
Usually costs: 1 mark.
Mistake: Forgetting units in growth rates
Writing r without per-day.
Do this: State the unit (per day or cm per day).
Usually costs: 1 mark.

Exam tips

Draw the sigmoid curve and label lag, log and stationary phases.
State the equation before substituting.
Give examples for every term.
Use the same time unit for r and t.

Practice questions on Growth Phases, Growth Rates and Measurement

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
A plant organ that shows an increase in length due to cells enlarging is in the:
(a) meristematic phase
(b) elongation phase
(c) maturation phase
(d) stationary phase
Hint
Vacuolation drives enlargement.
Answer
(b) elongation phase
Q2 · 2 marks · Short answerBoard level
What is heterophylly? Give an example.
Hint
Different leaf forms on one plant.
Answer
The occurrence of different shapes of leaves in the same plant, depending on age or environment; cotton and coriander (juvenile and mature leaves differ).
Q3 · 3 marks · Short answerHarder board level
Define plasticity in plants with an example.
Hint
Think about different forms.
Answer
The ability of plants to follow different pathways in response to environment or phases of life and form different structures. Example: the heterophylly of cotton, larkspur and coriander.
Q4 · 3 marks · Short answerHarder board level
An auxanometer has a pointer arm of 24 cm and a pulley of radius 2 cm. The pointer moves 30 mm. Find the actual growth of the plant.
Hint
Magnification is the ratio of the two radii.
Answer
Magnification = 24/2 = 12. Growth = 30 mm/12 = 2.5 mm.
Q5 · 3 marks · Assertion-reasonHarder board level
Assertion (A): Growth in plants is an irreversible process. Reason (R): It involves a permanent increase in size, mainly through cell division and enlargement. Choose: (a) both true and R explains A (b) both true, R does not explain A (c) A true, R false (d) A false, R true.
Hint
Define growth.
Answer
(a) Both true and R explains A.
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a student germinates gram and castor seeds.
(i) Name the type of germination in gram. (1 mark)
(ii) Name the type in castor. (1 mark)
(iii) Name the instrument used to measure growth indirectly. (1 mark)
(iv) State two conditions necessary for growth. (1 mark)
Hint
Recall the examples.
Answer
(i) Hypogeal. (ii) Epigeal. (iii) The auxanometer. (iv) Water, oxygen, nutrients (any two).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Equation of arithmetic growth?Show answer
Lₜ = L₀ + rt.
2.Instrument for indirect growth measurement?Show answer
Auxanometer.
3.Germination in which cotyledons stay underground?Show answer
Hypogeal.
4.Growth phases?Show answer
Meristematic, elongation, maturation.

Growth Phases, Growth Rates and Measurement: frequently asked questions

What is the difference between growth and development?

Growth is the increase in size; development includes growth, differentiation and the changes through the whole life cycle.

Why is growth in plants called open?

Because meristems remain active throughout life, so growth continues indefinitely.

What is the significance of dedifferentiation?

It allows differentiated cells to divide again, forming cambia that produce secondary tissues.