Investigating Resonance in Springs and Pendulums

Resonance is one of the most fascinating concepts in physics — when a system vibrates with maximum amplitude because it is driven at its natural frequency. Using springs and pendulums, students can observe resonance directly and understand why it is both useful and potentially destructive in the real world.

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The Experiment

Students set up a mass-spring system and a simple pendulum, each free to oscillate. A driver system (a mechanical vibrator or small motor) applies periodic forces at different frequencies. Lascells make a fantastic model for this, which is set up such that the strings are not tangled, and the experimental setup is immediately ready to go.

As the driving frequency changes, the amplitude of oscillation varies:

• At low or high frequencies, motion is small.

• At the natural frequency, amplitude increases dramatically — this is resonance.

The same can be shown using multiple pendulums of different lengths coupled by a thread; when one is set swinging, only the pendulum with the same natural frequency begins to move significantly.

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The Science

Resonance occurs when the frequency of a driving force matches the system’s natural frequency. Energy transfer is most efficient at this point, leading to a large increase in amplitude.

Key relationships:

$$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \quad \text{for a spring}$$ $$f = \frac{1}{2\pi}\sqrt{\frac{g}{l}} \quad \text{for a pendulum}$$

These equations show that the frequency depends on mass (for springs) and length (for pendulums).

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Skills Highlight

• Measuring oscillation frequency using timers or counting

• Plotting amplitude against driving frequency to identify resonance peaks

• Applying formulae to predict natural frequencies

• Understanding the practical implications of resonance in bridges, buildings, and musical instruments

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Why It Works in Teaching

Resonance links theory to experience — students can feel, hear, and see it. The rising amplitude at resonance provides an immediate visual and physical demonstration of a key principle of oscillatory motion, while the equations connect it back to quantitative analysis.

 

 

Investigating Leaf Pigments Using

Chromatography

Photosynthesis depends on more than just chlorophyll. Leaves contain a mixture of pigments that absorb different wavelengths of light. Using chromatography, students can separate these pigments and see the hidden colours that power photosynthesis.

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The Experiment

Students grind up a fresh green leaf with a small amount of ethanol or propanone to extract the pigments. A strip of chromatography paper is dipped into the solution, ensuring the pigment spot stays above the solvent level. As the solvent travels up the paper, it carries the pigments at different speeds, separating them into distinct bands.

Typical pigments seen include:

• Chlorophyll a – blue-green

• Chlorophyll b – yellow-green

• Xanthophylls – yellow

• Carotenes – orange

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The Science

Each pigment has a different solubility in the solvent and a different attraction to the paper. The more soluble pigments travel further, while less soluble ones remain near the baseline. This demonstrates the principle of separation by differential solubility.

Students can measure the Rf value for each pigment using:

$$Rf = \frac{\text{distance moved by pigment}}{\text{distance moved by solvent front}}$$

These values help identify unknown pigments and link to photosynthetic efficiency under different light conditions.

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Skills Highlight

• Applying chromatography to biological molecules

• Measuring and comparing Rf values

• Understanding pigment roles in light absorption

• Linking results to plant adaptation and light capture

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Why It Works in Teaching

Chromatography makes an invisible process visible. Students not only see that leaves contain more than one pigment but also learn how separation techniques reveal the complexity of photosynthesis. It is a simple, colourful experiment that connects molecular biology with plant physiology.

 

Supply and Demand – Why Prices Rise and Fall

Every time we buy something, from food to fuel, we see the laws of supply and demand in action. Understanding how these forces interact explains why prices rise, fall, or stabilise — and why markets behave the way they do.

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The Basics

Demand means how much of a product consumers want to buy at different prices.
Supply means how much producers are willing to sell at those prices.

• When prices fall, consumers buy more.

• When prices rise, producers are more willing to supply.

The point where the two meet is called market equilibrium — the price and quantity where supply equals demand.

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When Prices Change

Prices rarely stay at equilibrium for long.

• If demand rises (for example, due to a trend or shortage), prices increase until supply catches up.

• If supply rises (such as a bumper harvest or new technology), prices tend to fall.

• If demand falls (less interest or lower incomes), producers may cut prices to encourage sales.

These shifts constantly reshape markets, from housing and energy to concert tickets and video game consoles.

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Example: Fuel Prices

When oil supply falls because of production cuts or disruption, prices rise globally. When new sources or lower demand appear, prices drop. The same logic applies on a smaller scale to everyday goods — even coffee or avocados.

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Skills Highlight

• Understanding real-world data through graphs of supply and demand curves.

• Analysing market equilibrium and the effect of changes in supply or demand.

• Linking economic theory to current events and consumer behaviour.

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Why It Works in Teaching

Supply and demand gives students a clear way to connect theory with daily life. Whether it’s the cost of energy, food, or streaming subscriptions, they learn to think critically about why prices change and who gains or loses when they do.

 

Cybersecurity Basics – Cracking Simple Ciphers

Understanding cybersecurity begins with understanding how information can be protected — and how it can be broken. In the classroom, simple ciphers offer an engaging way to introduce encryption, decryption, and the logic behind keeping data safe.

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What is a Cipher?

A cipher is a method of changing a message so that only someone with the key can read it. The message before encryption is called plaintext, and after encryption it becomes ciphertext.

For example, using a Caesar cipher, each letter is shifted by a fixed number of places.

• Shift by 3: A → D, B → E, C → F
  So the message “HELLO” becomes “KHOOR”.

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Cracking the Code

Students can attempt to decrypt messages by:

• Trying different shift values (a brute-force approach).

• Looking for common letters or patterns such as “E” or “THE”.

• Comparing frequency counts of letters in the ciphertext with normal English usage.

This develops logical thinking and introduces real principles of codebreaking.

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Beyond Caesar

Once students understand simple substitution ciphers, they can explore:

• Keyword ciphers – where a secret word defines the letter substitutions.

• Transposition ciphers – where letters are rearranged rather than replaced.

• Frequency analysis – a statistical approach used in classical cryptography.

These challenges demonstrate that security relies not only on clever systems, but also on their resistance to reverse-engineering.

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Cybersecurity Connections

Modern encryption algorithms work on the same principles but at a vastly more complex scale. Understanding basic ciphers helps students see:

• Why strong passwords and encryption keys matter.

• Data protection relies heavily on mathematics and computation.

• Why cybersecurity is an essential modern skill.

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Why It Works in Teaching

Breaking codes gives students immediate feedback — a satisfying sense of discovery. It brings computing, logic, and maths together while encouraging persistence and reasoning.

Simple ciphers are a fun introduction to a serious subject: the protection of digital information in the modern world.

How to approach cracking a transposition cipher

1. Identify it’s a transposition cipher

Signs that a ciphertext is transposition (not substitution):

• Letter frequency looks like plain English (E, T, A still common) but text is unreadable.

• Common digrams/trigrams exist but in wrong positions.

• Ciphertext length unchanged; no new symbols introduced.

If letter frequencies are very altered, you may have a substitution or a mixed cipher.

2. Try to recognise the type

Common kinds you will meet in classroom puzzles:

• Columnar transposition (write plaintext in rows and read out by columns in some key order).

• Rail fence (zig-zag writing and read across rails).

• Route/spiral transpositions (write into a grid by route, read out by another route).

• Double transposition (apply two different columnar transpositions).

Start with columnar and rail fence — they’re the most frequent in exercises.

3. Useful observations and tests

• Look for probable words (cribs). If you suspect a word like THE, AND, or a place name, try to place it in candidates.

• Try likely key lengths. For columnar, reasonable classroom keys are small (3–10). For rail fence, try 2–6 rails.

• Examine repeated patterns. If the same group of letters recurs at regular intervals that suggests grid/column structure.

• Index of coincidence is usually close to English for transposition — use that to rule out substitution.

4. Manual columnar-cracking technique (basic)

For columnar transposition:

1. Guess a number of columns n. (If ciphertext length L, rows ≈ L/n.)

2. Write the ciphertext into n columns top-to-bottom (or into rows and then read by columns depending on convention).

3. Try permutations of column order to see which permutation yields readable plaintext.

4. Shortcuts:

   • If you see short common words split across columns, try permutations that bring those letters together.

   • Use likely cribs: place the crib in all possible positions in the candidate plaintext and see if columns can be re-ordered to make it match.

Manual solving is feasible for small n and short messages. For larger keys or double transpositions, use a computer.

5. Rail-fence cracking (quick)

• Try various rail counts r.

• Reconstruct plaintext by writing characters along a zig-zag of r rails and reading row by row; compare resulting text to English.

• A common clue is that small r values often reveal partial words quickly.

6. Automated / algorithmic approaches

When manual methods stall, use automated search:

• Brute force all key permutations (only feasible for small key lengths — n! grows fast).

• Scoring function: rate candidate plaintexts by English-language fitness (word-list hits, common bigrams/trigrams, log probability). Choose top-scoring candidates.

• Heuristic search: hill-climbing, simulated annealing or genetic algorithms can efficiently find good permutations for large keys (used in classical cryptanalysis).

• Crib dragging: if you know or guess a plaintext fragment, slide it along and check consistency with a transposition model.

7. Double transposition

• Try single transposition first to see if you get English fragments.

• If it’s double, you may need heuristic searches (hill-climbing) that try pairs of permutations and score results.

8. Practical classroom tips

• Teach students to start small: try rail fence and small column keys first.

• Use frequency tests to decide whether you should be looking for transposition or substitution.

• Give cribs (a known word) to classes — this shows how a tiny piece of plaintext can break many ciphers.

• Show comparison: same ciphertext, different decryptions — scoring separates readable English from junk.

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Small Python example: brute force columnar transposition (small keys)

This is a simple brute-force tester for a columnar transposition cipher. It tries key lengths up to 8 and all permutations for each length and prints candidates that contain a common English word (e.g. " the ").

Warning: permutations grow factorially. This script is fine for keys ≤ 8 but will be very slow above that.

# Simple brute-force columnar transposition cracker (educational)
from itertools import permutations
import math

def decrypt_columnar(cipher, key_order):
    n = len(key_order)
    L = len(cipher)
    rows = math.ceil(L / n)
    # compute number of "full" columns (some columns may be shorter by 1)
    cols_lengths = [rows] * n
    short_cols = (rows * n) - L
    if short_cols > 0:
        # last 'short_cols' columns will have one less char
        for i in range(n - short_cols, n):
            cols_lengths[i] -= 1
    # split cipher into columns in order (as written out by column)
    cols = []
    idx = 0
    for clen in cols_lengths:
        cols.append(cipher[idx: idx+clen])
        idx += clen
    # reorder columns according to key_order (key_order maps plaintext col pos -> cipher column index)
    # here key_order is a tuple of indices representing the order columns were read; we want to reconstruct rows
    # We need to place cols back into their plaintext column positions
    plain_cols = [None] * n
    for plain_pos, col_idx in enumerate(key_order):
        plain_cols[plain_pos] = cols[col_idx]
    # read off row-wise
    plaintext = []
    for r in range(rows):
        for c in range(n):
            if r < len(plain_cols[c]):
                plaintext.append(plain_cols[c][r])
    return ''.join(plaintext)

cipher = "WEEENNHRREE T..."  # replace with your ciphertext (no spaces ideally)
cipher = ''.join(cipher.split())  # remove whitespace for processing

common_word = " the "  # crude filter
for n in range(2, 9):  # try key lengths 2..8
    print(f"Trying key length {n} (permutations: {math.factorial(n)})")
    for perm in permutations(range(n)):
        # perm describes the order columns were read out when encrypting
        pt = decrypt_columnar(cipher, perm)
        if common_word in (' ' + pt.lower() + ' '):
            print("Possible plaintext (key order):", perm)
            print(pt[:200])

This script is a classroom starter — you can improve it by:

• Using a better scoring function (word frequency / log probability).

• Allowing for different conventions (writing rows then reading columns, or vice versa).

• Adding support for double transposition (nested search or heuristic search).

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Example of heuristic scoring (short note)

A robust solver uses a scoring function based on English quadgram or trigram frequencies. The algorithm randomly permutes the key order and uses hill-climbing: try small swaps in the permutation and keep changes that improve the score. Repeat until no improvement — repeat from different random starts. This is how many classical puzzle solvers succeed at larger keys or double transpositions.

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Common pitfalls

• Forgetting encryption convention: whether plaintext was written in rows and read by columns, or vice versa. Try both conventions.

• Not accounting for short columns when length L is not divisible by key length.

• Assuming a single transposition when it's double.

• Trying brute force for large keys will result in an exponential combinatorial explosion.

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Quick classroom exercise

Give students a short plaintext (30–50 letters), apply a small columnar key (3–5), and then challenge classmates to recover it using:

• frequency check,

• guessing key length,

• manual column re-ordering,

• cribbing.

Then show how a small program can solve it in seconds.

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Worksheet: Cracking Transposition Ciphers

1. Warm-up: Rail-Fence Cipher

In a rail-fence cipher, the message is written diagonally across a set number of “rails”, then read row by row.

For example, with 3 rails:

HELLO WORLD
H . . . O . . . R . .
. E . L . W . L . D .
. . L . . . O . . . .

The ciphertext would be: H O R E L W L D L O

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Cipher 1

Ciphertext:

WECRLTEERDSOEEFEAOCAIVDEN

1. Try different rail counts (2, 3, 4, 5).

2. Write the letters in a zig-zag pattern for each rail count.

3. Which number of rails produces an English-like plaintext?

4. Write your decoded message below:

Decrypted text: ____________________________________________

(Hint: The correct answer has three rails.)

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2. Columnar Transposition Cipher

In a columnar cipher, the message is written in rows under a keyword and then read column by column in the order of the keyword’s letters.

Example with keyword CAGE (alphabetical order 1–4):

C(2)	A(1)	G(3)	E(4)
H	E	L	L
O	W	O	R
L	D

Ciphertext (columns in order 1-2-3-4): EWDHLOOLR

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Cipher 2

Ciphertext:

TIEHXHETATSRHESAESPNLTTE

Known information:

• The keyword is SALT (4 letters).

• The plaintext is an English sentence.

1. Write the ciphertext in 4 columns, one column per keyword letter.

2. Work out which column comes first, second, third, and fourth according to the alphabetical order of the keyword letters.

3. Read the message row by row to recover the plaintext.

4. Write your decoded sentence below:

Decrypted text: ____________________________________________

(Hint: You should find a sentence about how salt preserves food.)

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3. Reflection Questions

1. How can you tell that both ciphers use transposition rather than substitution?

2. Which cipher was easier to break, and why?

3. How might computers speed up this process for longer messages?

4. Why are transposition ciphers no longer secure for modern communication?

 

Electrolysis of Solutions – Splitting Water and Beyond

Electrolysis is one of the most dramatic chemistry experiments. It turns electricity into chemistry by decomposing compounds into their elements. With a Hoffman voltameter, students can see this process in action — as bubbles of hydrogen and oxygen gas appear right before their eyes.

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The Setup

A Hoffman voltameter consists of three vertical glass tubes connected at the bottom and filled with a conductive solution, usually acidified water.

• Electrodes are connected to a direct current power supply.

• When current flows, hydrogen gas forms at the negative electrode (cathode) and oxygen gas at the positive electrode (anode).

• The gases collect in separate graduated tubes, making the reaction measurable and visible.

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The Reaction

$$2H_2O(l) \rightarrow 2H_2(g) + O_2(g)$$

The key observation is the 2:1 ratio of hydrogen to oxygen — twice as much hydrogen gas forms because water contains two hydrogen atoms for every oxygen atom.

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Measuring Results

Students can measure the gas volumes and compare them to the theoretical ratio. Typical results show:

Gas	Electrode	Volume Collected (cm³)
Hydrogen	Cathode	20
Oxygen	Anode	10

When tested, hydrogen burns with a squeaky pop, while oxygen relights a glowing splint — confirming their identities.

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Beyond Water

The same principles apply to other electrolytes:

• Copper(II) sulfate solution: copper forms on the cathode, oxygen at the anode.

• Sodium chloride solution: hydrogen at the cathode, chlorine gas at the anode.

By changing the electrolyte and electrodes, students explore how ion discharge depends on reactivity and concentration.

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Skills Highlight

• Writing ionic equations for electrolysis.

• Identifying products at each electrode.

• Measuring and interpreting experimental gas ratios.

• Linking practical results to energy and bonding.

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Why It Works in Teaching

Electrolysis is one of those experiments that bridges chemistry and physics. The Hoffman voltameter makes an invisible process visible, turning abstract equations into real, testable phenomena. Students don’t just learn what electrolysis means — they see it, measure it, and prove it.

 

Investigating Electricity With PASCO Current Sensors

Understanding electricity is easier when students can measure it directly. PASCO current sensors make invisible electrical quantities visible, allowing learners to explore current flow, Ohm’s Law, and circuit behaviour with real-time data.

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The Setup

A PASCO current sensor connects easily into a circuit, just like an ammeter. When linked to PASCO software or a data logger, it records current continuously and displays it on a graph.

Typical investigations include:

• Measuring how current changes with voltage in a simple circuit.

• Comparing current in series and parallel circuits.

• Observing how resistance affects current flow.

The advantage is accuracy — and instant visual feedback.

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Experiments in Action

1. Ohm’s Law
   Students vary the potential difference across a resistor and record the current. Plotting current against voltage produces a straight line through the origin, confirming that $V = IR$.

2. Series and Parallel Circuits
   By building and measuring different circuit types, students discover that:

• In series circuits, current is the same everywhere.

• In parallel circuits, current splits between branches.

3. Temperature and Resistance
   Using a filament bulb or thermistor, students can measure how resistance changes as the component warms up.

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Skills Highlight

• Accurate measurement and graph plotting.

• Understanding relationships between current, voltage, and resistance.

• Designing and modifying circuits safely.

• Analysing real data instead of relying on theoretical values.

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Why It Works in Teaching

The PASCO current sensor turns theory into observation. Students see the link between voltage and current unfold on screen in real time. It builds data-handling skills, reinforces key electrical relationships, and shows how digital sensors improve accuracy over traditional meters.

Electricity becomes a measurable, dynamic process rather than a set of abstract equations.

 

Trigonometry in the Real World – Measuring Heights With Shadows

Trigonometry is often introduced with triangles drawn on paper, but its real power comes when students take it outside. One of the simplest and most satisfying applications is using shadows to measure the height of tall objects — from trees and lampposts to buildings.

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The Principle

When sunlight hits an object, it forms a right-angled triangle between the object, its shadow, and the line of sight to the top.
If students measure:

• The length of the shadow, and

• The angle of elevation from the tip of the shadow to the top of the object,
  they can use trigonometry to find the object’s height.

Using:

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ $$\text{height} = \text{shadow length} \times \tan(\theta)$$

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Example Calculation

Object	Shadow Length (m)	Angle of Elevation (°)	Calculated Height (m)
Tree	5.0	35	3.5
Lamp Post	7.5	40	6.3
Building	10.0	45	10.0

Students can verify results by comparing with a tape measure or building data, reinforcing accuracy and proportional reasoning.

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Skills Highlight

• Applying trigonometric ratios (tan, sin, cos) to real situations.

• Measuring and estimating angles with a clinometer or phone app.

• Recognising sources of error such as uneven ground or moving shadows.

• Relating maths to everyday objects and outdoor measurement.

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Why It Works in Teaching

Taking trigonometry outdoors turns numbers into meaning. Students see how angles and ratios describe the real world. It transforms abstract formulas into a practical tool for problem-solving — and shows that maths genuinely measures the world around us.

Measuring Height Using Shadows – Trigonometry in Action

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Layout Description

1. Scene Illustration (Top Section)

• A simple side-on diagram showing:

  • A tree or lamppost standing vertically on the left.

  • A horizontal ground line extending to the right.

  • A shadow cast along the ground (labelled shadow length = adjacent side).

  • A ray of sunlight coming down diagonally to the tip of the shadow, forming a right-angled triangle.

  • The angle of elevation (θ) marked between the shadow and the line to the top of the object.

Labels:

• Height of object (opposite side)

• Shadow length (adjacent side)

• Angle θ (angle of elevation)

• Sunlight direction

• Right angle marked between the height and the ground.

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2. Formula Section (Middle)
Clearly display the key equation underneath the diagram:

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \quad \Rightarrow \quad \text{Height} = \text{Shadow Length} \times \tan(\theta)$$

Below this, show a quick worked example:

Example: Shadow = 7.5 m, Angle = 40°

Height = 7.5 × tan(40°) = 6.3 m

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3. Quick Tips (Bottom Section)
Three short side boxes:

• Tip 1: Measure the shadow quickly before the sun moves.

• Tip 2: Use a clinometer or phone app to measure the angle.

• Tip 3: Keep the ground level for accurate results.