Why a Can Opener Is a Small Machine-Laboratory
A manual can opener turns a comfortable hand motion into a controlled cut around a metal rim. Inside that everyday action are gears, circles, torque, friction, path length and a trade-off between force and distance. This is a cheerful example of why mathematics matters: the numbers explain not only how far a wheel moves, but why a longer handle may feel easier and why the cutter must remain aligned with the rim.
The mechanism should be studied through drawings, clean cardboard models or an unused display tool. A real lid can be dangerously sharp. A classroom calculation explains a mechanism; it does not certify that a particular opener or cut edge is safe.
The Mechanism in One Picture
Handle, gears, feed wheel and cutter
Many rotating can openers use a handle or crank connected to a gear. That gear drives a smaller or larger gear attached to a toothed feed wheel. The feed wheel grips the can rim while a cutting wheel pierces or shears the lid. Patent descriptions show the feed wheel advancing the can relative to the cutter, and some designs use intermeshing gears to coordinate the parts: see the rotary feed-wheel patent record and a gear-engaging side-cut design.
The exact arrangement differs by model. That is a useful mathematical warning: draw the mechanism you actually observe before assigning driver and driven gears.
Gear ratio
If a 36-tooth driving gear turns a 12-tooth driven gear, the driven gear makes 36/12 = 3 turns for each driver turn, ignoring slip. Its angular speed is three times as large, while its ideal torque is one-third as large. If the 12-tooth gear drives the 36-tooth gear, the result reverses: one input turn gives one-third of an output turn and, ideally, three times the torque.
| Driver teeth | Driven teeth | Output turns per input turn | Ideal torque pattern |
|---|---|---|---|
| 36 | 12 | 3 | faster, less output torque |
| 12 | 36 | 1/3 | slower, more output torque |
Real mechanisms lose energy through friction, deformation and slip. A gear ratio predicts a kinematic relationship, not perfect efficiency.
Circular Paths and Feed Distance
Circumference turns rotation into travel
If a feed wheel of radius 8 mm rolls without slipping, one revolution advances approximately 2πr = 16π mm, or 50.3 mm. Ten ideal revolutions would advance about 503 mm. That distance may not equal the cutter's exact path because the effective contact radius, rim geometry and slip matter.
For a circular lid of radius 42 mm, the required path around the rim is approximately 2π(42) = 263.9 mm. Dividing by 50.3 mm suggests about 5.25 feed-wheel revolutions in an ideal model. A partial revolution at the end is perfectly sensible; the mathematical path does not have to contain a whole number of turns.
Did You Know? Changing the feed-wheel radius changes distance per revolution. It may also change contact forces, tooth engagement and packaging space, so one “bigger is better” rule would be misleading.
Angle and arc length
Arc length can be written s = rθ when θ is in radians. If the can advances through 90 degrees, θ = π/2 and a cutter path at radius 42 mm has length 42π/2 ≈ 66.0 mm. This representation makes partial cuts easy to compare.
Mechanical Advantage and Torque
Torque is force multiplied by perpendicular lever arm. If a student applies 18 N at a crank radius of 45 mm, the input torque is 18 × 0.045 = 0.81 N·m. Applying the same force at 30 mm gives 0.54 N·m. The longer crank produces 50% more torque, although the hand must travel farther in each turn.
That trade-off is central to machines. A larger mechanical advantage can reduce required force, but it increases travel or reduces speed. It does not create energy. If the hand moves through a circle of radius 45 mm, one revolution covers about 283 mm; at 30 mm it covers about 188 mm.
A combined worked example
Suppose the crank gear has 24 teeth and drives a 12-tooth gear on a feed wheel. One crank turn produces two feed-wheel turns. If the feed wheel's effective radius is 7 mm, ideal advance per crank turn is 2 × 2π × 7 ≈ 88.0 mm. A 280 mm cutting path therefore needs about 280/88.0 = 3.18 crank turns.
Now allow 8% slip. Effective advance becomes 0.92 × 88.0 ≈ 81.0 mm, and the estimate rises to about 3.46 turns. Slip changes the observed count without changing the printed tooth numbers. This is why a physical test and a mathematical model belong together.
Measurement, Error and Honest Conclusions
A photograph can support measurements of tooth count and diameter, but perspective distorts lengths. A ruler on the same plane as the mechanism provides a scale. Repeat each measurement and report a sensible resolution.
If the feed-wheel radius is 7.0 ± 0.2 mm, circumference is about 44.0 mm and changes by about 2π(0.2) ≈ 1.26 mm across the stated radius uncertainty. Multiplying by gear ratio and turns propagates that uncertainty. Extra decimal places do not make the physical radius exact.
Backlash is another useful idea. Reversing the handle may rotate a small angle before the driven gear moves because clearance exists between teeth. That lost motion matters when direction changes, even if it is unimportant during steady forward cutting.
Common Misconceptions
“More gear teeth always means more power”
Tooth count matters relative to the mating gear. Ratio changes speed and torque; material, tooth shape, friction and strength also matter.
“The cutter travels exactly one lid circumference”
The effective cut radius may differ from the can's labelled radius, and entry, overlap or side-cut geometry can add or subtract path.
“Mechanical advantage makes free energy”
An ideal machine exchanges force for distance. Real machines also lose some input energy.
“A neat calculation proves the edge is safe”
It does not. Edge condition is an observed safety issue, not a consequence of a gear-ratio equation.
How Students Can Learn the Mathematics
- Build two meshing cardboard gears and mark one tooth on each.
- Predict turns using tooth counts, then test the prediction.
- Draw the can rim and calculate a full and partial arc.
- Compare torque for two safe model-handle lengths.
- Record uncertainty in radius and repeat the calculation.
- Explain the result with both a diagram and a sentence about limitations.
Parents can ask, “Which number comes from geometry, and which comes from observation?” That question separates facts about the model from facts about the particular tool.
Useful Next Reading
For ordinary kitchen technique, read How to Master Using a Can Opener. For broader mathematical connections, continue through the eduKate Mathematics Learning Hub.
Frequently Asked Questions
Why does a can opener need gears?
Gears can transmit rotation, coordinate the feed and cutter, and trade speed for torque according to their ratio.
How many turns should open a can?
It depends on lid path length, feed-wheel effective radius, gear ratio, engagement and slip. The worked model shows how to estimate it, not a universal count.
Is torque the same as force?
No. Torque depends on force and its perpendicular distance from the rotation axis.
What is the fairest conclusion?
State what the chosen geometry predicts, compare it with a safe model or manufacturer information, and name friction, slip and measurement uncertainty.
The Bigger Answer to “Why Mathematics?”
A can opener shows how mathematics connects movement to purpose. Ratios coordinate gears, circumference converts turns into travel, torque explains leverage and uncertainty keeps the answer honest. The result is not merely a number of turns; it is a clear explanation of how a small machine transforms a hand motion into a controlled path.
A Practical Mathematics Studio
Use diagrams, cardboard gears or a clean unplugged display tool. Do not dismantle a powered opener or handle a freshly cut lid; sharp metal edges and pinch points make real opening an adult kitchen task. Keep raw observations, units, assumptions and limitations beside each result.
Investigation 1: Map the handle-to-wheel mechanism
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 2: Count gear teeth from a drawing
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 3: Calculate a gear ratio
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 4: Relate handle turns to feed-wheel turns
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 5: Measure a model handle radius
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 6: Compare input and output torque
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 7: Draw the rim contact forces
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 8: Estimate feed distance per revolution
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 9: Calculate a circular lid path
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 10: Compare lid and feed-wheel circumference
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 11: Model cutter penetration depth
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 12: Separate speed from mechanical advantage
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 13: Track angular displacement
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 14: Estimate work over one turn
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 15: Compare two handle lengths
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 16: Test a cardboard gear model
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 17: Plot advance against turns
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 18: Check for tooth slip
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 19: Identify backlash
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 20: Audit diameter versus radius
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 21: Run a sensitivity check
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 22: Estimate measurement uncertainty
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 23: Compare top-cut and side-cut paths
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 24: Inspect a partial final segment
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 25: Recalculate with exact pi
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 26: Round only at the end
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 27: Write a sharp-edge boundary
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 28: Separate model from product rating
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 29: Explain one failed assumption
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Investigation 30: State a bounded conclusion
Declare every radius, tooth count, angle and revolution used. Draw the relevant gear or circular path, calculate the named ratio, distance, torque or work quantity, and show how one assumption changes the result.
Evidence check: Check gear engagement, rotation direction and whether the cutter or can is advancing. Mechanical advantage trades distance or speed for force and does not create energy. Recalculate one case independently and compare the result with a graph, diagram, table or second run before drawing a conclusion.
Boundary: The exercise cannot certify a can opener, guarantee a smooth edge or make a damaged tool safe. Follow product instructions and normal food and sharp-edge precautions. End with one sentence stating what the mathematics supports and one naming the evidence still missing.
Studio Synthesis
Choose four investigations that use different representations: a labelled system diagram, an equation, a raw-data table and a graph. Arrange the report from question to method, calculation, residual or sensitivity check, limitations and conclusion. Another reader should be able to reproduce one result without guessing a unit, category or selection rule.
Change one input or classification rule at a time and recalculate. Identify which conclusion remains stable and which changes direction. Sensitivity analysis reveals the assumptions controlling a result and is usually more valuable than an extra decimal place.
Exchange only the raw data and definitions with a partner. If the calculation reproduces and the same bounded conclusion follows, confidence increases. If it does not, repair the method or labels before treating the output as evidence.