Inquiry Project: No Ladder Required | Geometry Unit 2 | Welch Teaching & Coaching Website

The Driving Question

Answer this by the end of the project

How tall is it — and how sure are you? Measure something you cannot climb, cannot touch the top of, and cannot put a tape measure against.

No ladder. No drone. No phone app that claims to do it for you. Cardboard, string, a mirror, a stick, and Unit 2.

Every method in this project rests on the same idea: if you can build a triangle that is similar to the big invisible triangle formed by you, the ground, and the top of the object, then the ratio you can measure equals the ratio you cannot. That is the whole of Lesson 3, put to work outdoors. The wedge in Phase 0 is built on Lesson 2, the sighting stick in Phase 4 is the side-splitter theorem from Lesson 4, and Phase 5 is Lesson 5 with two pieces of string.

The point of the project is not to get a number. It is to get five numbers that disagree, and then to reason your way to one answer you are willing to write down and defend.

Watch at least one of these before the build day. They show the same ideas being used on real objects. Videos do not appear on the printed copy.

Applications — indirect measurement Similar triangle application: measuring across a gorge Patrick Sullivan Open the video →

Applications — heights you cannot reach Application of similar triangles rodcastmath Open the video →

Applications — the mirror idea Triangle similarity in pool Khan Academy Open the video →

Applications — scale factor outdoors Scale and indirect measurement Khan Academy Open the video →

What you will turn in

1. The wedge

A cardboard \(45^\circ\) sighting wedge with a plumb line, plus a written justification of why its two acute angles must be \(45^\circ\).

2. Five labeled diagrams

One per method, each with a similarity statement and the proportion you actually solved.

3. The data table

Every raw measurement, in the units you took it in, with the person who took it named.

4. Five computed heights

Worked arithmetic for each method, not just the answer, rounded to two decimal places.

5. The comparison

Percent difference between every method and the group mean, and an identified outlier with a physical explanation.

6. The defense

One final height with an error bar, plus a three-minute group argument for why that is the number to trust.

Phase 0 — Build the Tool

P0

The \(45^\circ\) sighting wedge

Half a class period • indoors

Cut a right triangle out of stiff cardboard with the two legs the same length. Do not measure the angles. Do not use a protractor. Measure the legs, and let the geometry deliver the angles for free.

The two legs are cut to the same length, so the tool is a right isosceles triangle and both of the other angles must be \(45^\circ\). That is the Isosceles Triangle Theorem doing real work: you never measure the angles, you guarantee them by making the legs congruent.

Do this

Pick a leg length and write it down. Anything from 8 to 12 inches works; longer is steadier to sight along, shorter is easier to hold level.
Mark both legs to exactly that length from the same corner, using the same ruler and the same starting mark. Equal legs are the entire mathematical content of the tool — if they are not equal, nothing downstream is valid.
Cut along the hypotenuse. Keep the cut straight; that edge is your sight line.
Tape a string with a small weight (a washer, a nut, a key) at the corner where the two legs meet, so it hangs freely along one leg. That is your plumb line: when the string lies flat against the leg, the other leg is level.
Measure both legs one more time with a different ruler and record the actual lengths. Any difference between them is your first known source of error, and you will refer back to it in Phase 6.

Justify it

Write this argument on the front of your packet, in your own words. It is graded.

The triangle has a right angle, so the other two angles must total \(90^\circ\). The two legs were cut congruent, so by the Isosceles Triangle Theorem the two angles opposite them are congruent. Two congruent angles adding to \(90^\circ\) leaves only one possibility:

\[\text{each acute angle}=\frac{180-90}{2}=45^\circ\]

Why this matters. You never measured a \(45^\circ\) angle, and you do not have to trust that you cut one. You measured two lengths, which is something a ruler does well, and the theorem converted them into an angle guarantee. A protractor gives you an angle you hope is right. The Isosceles Triangle Theorem gives you an angle that cannot be anything else.

Questions to answer in writing

Suppose one leg came out \(3\%\) longer than the other. Are the two acute angles still congruent? What does the theorem still guarantee, and what does it no longer guarantee?
Why does the plumb line matter? Describe what goes wrong in Phase 1 if the bottom leg is tilted up by a few degrees.

Phase 1 — Method 1: The \(45^\circ\) Sight

P1

Walk until the angle is right

Half a class period • outdoors

This is the only method where you do not solve a proportion at all. You walk backward until the geometry hands you the answer.

Back up until the top of the object sits exactly on the sighting edge. Your sight line, your eye-level line, and the object now form a \(45^\circ\)–\(45^\circ\)–\(90^\circ\) triangle, so the two legs are congruent: the height above your eye equals your distance out. Add your eye height and you have the whole thing.

Do this

Hold the wedge with the plumb line hanging flat against the vertical leg. Sight along the hypotenuse toward the top of the object.
Walk toward or away from the object until the very top of it sits exactly on your sight line. Hold still and have a partner mark the ground at your feet.
Measure the horizontal distance \(d\) from that mark to the base of the object. Measure along the ground, not along your sight line.
Measure your eye height \(e\) — from the ground to the corner of the wedge you were sighting from, not to the top of your head.
Repeat the whole thing with a different person sighting. Record both results separately. Do not average them yet.

The mathematics

Your sight line, the horizontal line at eye level, and the object form a triangle with a \(45^\circ\) angle at your eye and a \(90^\circ\) angle at the object. The third angle is therefore also \(45^\circ\), which makes the triangle isosceles by the converse of the Isosceles Triangle Theorem — and the two legs congruent.

\[H=d+e\]

The height above your eye equals your distance out. Add your eye height and you have the whole object. No proportion, no calculator.

The step everyone forgets. Adding \(e\) at the end. The triangle only reaches down to your eye, and a group that skips this line is consistently short by about five feet — which is exactly the kind of systematic error Phase 6 asks you to hunt for.

Questions to answer in writing

Two people in your group got different distances \(d\). List two physical reasons why, and say which one you think mattered more.
This method works without any arithmetic. What did you give up in exchange for that convenience?

Phase 2 — Method 2: Shadows

P2

Let the sun build the second triangle

Half a class period • outdoors, sunny

Stand a meter stick upright next to the object. The sun casts a shadow from both. Two triangles, one ratio.

The sun is far enough away that its rays strike both objects at the same angle, so the two marked angles are congruent and both triangles have a right angle at the ground. That is enough for similarity, which gives \(\tfrac{H}{S}=\tfrac{h}{s}\).

Do this

Stand the meter stick vertically on level ground near the object. Check vertical with the plumb line from your wedge, not by eye.
Measure the stick's shadow \(s\) from the base of the stick to the tip of the shadow.
Measure the object's shadow \(S\) the same way, from the base of the object to the tip. Do it immediately — within a minute or two of the stick reading.
Record the time of day next to both measurements.
Take a second pair of readings at least fifteen minutes later, and record those separately.

The mathematics

The sun is far enough away that its rays arrive at both objects at the same angle. Both objects stand square to the ground. Two pairs of congruent angles makes the triangles similar, and similar triangles have proportional corresponding sides.

\[\frac{H}{S}=\frac{h}{s}\;\Longrightarrow\;H=\frac{h\cdot S}{s}\]

Worked example. A \(1\) m stick casts a \(0.8\) m shadow. The object casts a \(26\) m shadow.

\[H=\frac{1\times 26}{0.8}=32.5\text{ m}\]

Units. Pick one unit for the whole method and stay in it. Mixing a stick measured in centimetres with a shadow measured in feet produces an answer that is wrong by a factor of thirty and looks perfectly reasonable on the page. Write the unit next to every number as you record it.

Questions to answer in writing

Your two sets of readings, fifteen minutes apart, give two different heights. They should not. Explain the discrepancy in terms of what changed and what did not.
What happens to this method on an overcast day, and why is that a limitation of the method rather than of your measuring?
If the ground slopes away from the object, is your measured shadow too long or too short? Which direction does that push your answer?

Phase 3 — Method 3: The Mirror

P3

Bounce a sight line off the ground

Half a class period • outdoors, any weather

A flat mirror on the ground turns your line of sight into a bent path with two similar triangles on either side of the bend.

Light leaves the mirror at the same angle it arrives, so \(\angle 1\cong\angle 2\). Both triangles stand square on the ground, so they are similar and \(\tfrac{H}{D}=\tfrac{e}{d}\).

Do this

Place the mirror flat on level ground, roughly \(20\) to \(40\) feet from the base of the object. Put a small piece of tape with an X on it at the centre of the mirror — that X is your target point.
Walk backward, watching the mirror, until the very top of the object appears exactly on the X.
Stop and stand still. Measure \(d\), from your toes to the X, and \(D\), from the X to the base of the object.
Measure your eye height \(e\) again — the same measurement as Phase 1, from the ground to your eye.
Move the mirror to a noticeably different distance and repeat the whole procedure. Record both results.

The mathematics

Light reflects off the mirror at the same angle it arrives, so the angle between your sight line and the ground equals the angle between the object's line and the ground. You and the object both stand perpendicular to the ground. Two pairs of congruent angles again gives similar triangles.

\[\frac{H}{D}=\frac{e}{d}\;\Longrightarrow\;H=\frac{e\cdot D}{d}\]

Worked example. Eye height \(e=5.2\) ft, \(d=3.5\) ft to the mirror, \(D=28\) ft from the mirror to the object.

\[H=\frac{5.2\times 28}{3.5}=41.6\text{ ft}\]

Set the mirror up honestly. The mirror must be flat and level, and you must move yourself until the top lands on the X — not slide the mirror until the reflection looks right while standing still. Sliding the mirror changes \(D\) and \(d\) at the same time and quietly breaks the proportion.

Questions to answer in writing

You measured with the mirror at two different distances. Compare the two heights. Which setup do you trust more, and what in your data supports that?
This method depends on a fact from physics, not geometry. Name it, and explain which pair of congruent angles it is responsible for.
Which single measurement in this method would hurt the most if it were off by one inch? Justify the choice using the proportion, not intuition.

Phase 4 — Method 4: The Sighting Stick

P4

Split the side of the big triangle

Half a class period • outdoors

This one is the side-splitter theorem from Lesson 4, standing upright in the grass.

The stick and the object are both vertical, so \(\overline{DE}\parallel\overline{BC}\) and the Triangle Proportionality Theorem applies directly: \(\tfrac{AD}{AB}=\tfrac{DE}{BC}\).

Do this

Lie down, or crouch, and put your eye at ground level at a fixed spot. Mark that spot — call it \(A\).
Have a partner hold the meter stick vertically between you and the object, plumb line checked.
Have your partner walk the stick toward or away from you until the top of the stick lines up exactly with the top of the object from your eye at \(A\).
Measure \(AD\), from your eye position to the base of the stick, and \(AB\), from your eye position to the base of the object.
Record the stick height \(DE\). Repeat with the stick at a different distance and record that result too.

The mathematics

The stick and the object are both vertical, so \(\overline{DE}\parallel\overline{BC}\). The Triangle Proportionality Theorem applies to the two sides that \(\overline{DE}\) crosses, and the same ratio governs the parallel sides:

\[\frac{AD}{AB}=\frac{DE}{BC}\;\Longrightarrow\;BC=\frac{DE\cdot AB}{AD}\]

Worked example. Stick height \(DE=1\) m, \(AD=2.4\) m, \(AB=78\) m.

\[BC=\frac{1\times 78}{2.4}=32.5\text{ m}\]

Piece over whole, on purpose. Note that \(AB\) is the entire distance from your eye to the object, not the leftover piece \(DB\). This is the distinction the coach note in Lesson 4 warns about, and here it is not an abstraction — using \(DB\) instead of \(AB\) will hand you a height that is wrong by tens of feet.

Questions to answer in writing

What would happen to your answer if the stick were tilted \(5^\circ\) toward you? Would the height come out too large or too small? Explain using the theorem.
Would this method be more accurate with the stick close to your eye or close to the object? Argue from your two trials, not from a guess.
Rewrite the proportion in the piece-over-piece form, using \(DB\) instead of \(AB\), and show that it gives the same height. What does \(\overline{DE}\) correspond to in that version?

Phase 5 — Method 5: The Geometric Mean

P5

Two strings and a right angle

Half a class period • outdoors

The last method uses Lesson 5 directly, and it is the only one that needs no ruler held vertically at all.

Pull two strings from the two ends of a measured baseline until they meet at a right angle. The height of that meeting point above the baseline is the geometric mean of the two pieces: \(h=\sqrt{AD\cdot DB}\).

Do this

Two group members stand on opposite sides of the object's base, in a straight line through it. Call the positions \(A\) and \(B\), and mark both.
Each runs a string from their ground position up to the top of the object. Pull them taut. A third member sights the corner where the two strings meet at the top.
Adjust one person's position until the two strings meet at a right angle at the top. Check the corner with your cardboard wedge held flat against it — the \(90^\circ\) corner of the wedge is exactly the tool for this.
Measure \(AD\) and \(DB\), the two ground distances from each person to the base of the object.
Record both distances and the total \(AB=AD+DB\).

The mathematics

With a right angle at the top, the object itself is the altitude drawn to the hypotenuse \(\overline{AB}\) of a right triangle, and the base splits that hypotenuse into \(AD\) and \(DB\). The altitude is the geometric mean of the two pieces:

\[H=\sqrt{AD\cdot DB}\]

Worked example. \(AD=18\) ft and \(DB=72\) ft.

\[H=\sqrt{18\times 72}=\sqrt{1296}=36\text{ ft}\]

Confirm it with the leg relationships if you want a second check: the two strings should measure \(\sqrt{18\cdot 90}\approx 40.25\) ft and \(\sqrt{72\cdot 90}\approx 80.50\) ft. If your strings are close to those numbers, the right angle at the top was real.

This is the fussiest method, and that is the point. Getting the corner to a true \(90^\circ\) is hard, and small errors in the angle move the answer a lot. Expect this one to be your outlier. An outlier you can explain is worth more in Phase 6 than a number that happened to land near the others.

Questions to answer in writing

String lengths let you check the right angle two ways: the geometric-mean leg relationships above, and the Pythagorean Theorem from Unit 1. Do both, and report whether they agree.
If the corner at the top was actually \(85^\circ\) rather than \(90^\circ\), would \(\sqrt{AD\cdot DB}\) overestimate or underestimate the height? Explain your reasoning.

Phase 6 — Defend One Number

P6

Five answers, one decision

One class period • indoors

You now have five heights for one object, and they disagree. This phase is the reason the project exists.

Do this

Fill in the comparison table below with all five heights, converted to a single common unit.
Compute the mean of the five. Then compute the percent difference of each method from that mean.
Identify your largest outlier. Name a specific physical cause — a leg that was \(2\) mm off, a shadow tip you could not see clearly, a mirror on sloped ground. “Human error” earns nothing.
Decide whether to throw the outlier out. Write the decision and the reason. Both keeping it and discarding it can be correct; only an unjustified choice is wrong.
State one final height with an error bar, in the form \(H=\underline{\quad}\pm\underline{\quad}\). Justify the size of the \(\pm\) from your own spread, not from a feeling.

Method	% difference from the mean	Trust: high / medium / low
1. \(45^\circ\) wedge
2. Shadows
3. Mirror
4. Sighting stick
5. Geometric mean
Mean of all five	—	—

\[\text{percent difference}=\frac{|\text{method}-\text{mean}|}{\text{mean}}\times 100\%\]

Questions to answer in writing

Which method do you trust most, and why? Your answer must cite something you observed while taking the measurements, not the fact that it agreed with the others.
Are your five results scattered randomly around the mean, or are most of them on one side of it? A one-sided spread means a systematic error — find it and name it.
Every method here reduces to one proportion. Write all five proportions in a single list. What do they have in common, and what is the one thing each of them needed you to guarantee?
Suppose you had to give this measurement to a contractor who was going to order material based on it. Would you report your mean, your most-trusted single method, or something else? Defend the choice.

The defense. Three minutes per group. Pick one decision — the outlier you discarded, the method you trusted, the size of your error bar — and argue it with your own numbers on the board. Every member speaks. The strongest defenses in this project are usually the ones that begin by admitting what went wrong.

Materials & Logistics

Per group of 3–4

A sheet of stiff cardboard, foam board, or corrugated plastic, at least 12 inches square
A ruler or yardstick, plus scissors or a utility knife (teacher-supervised)
String and a small weight for the plumb line, plus tape
One meter stick or yardstick for Phases 2 and 4
One small flat mirror, roughly 4 by 6 inches, with the edges taped
One 50-foot or 100-foot tape measure
Two lengths of non-stretch string, at least 100 feet each, for Phase 5
Chalk or marking flags, a clipboard, a calculator, and a phone or camera

Safety and courtesy

Never sight toward the sun, and never use the mirror to reflect sunlight at anyone. Tape the mirror edges before it leaves the room.
No climbing, no ladders, no leaning out of windows. The entire point is that the top stays out of reach.
Keep all measuring lines and strings off roads, driveways, and walkways. Post a spotter if a line must cross one.
Phase 4 has a student lying on the ground — choose a spot well away from traffic and mowing crews, and keep a group member standing.
Cutting is done at the table, indoors, with the blade pulled away from the body. Knives are counted out and counted back in.
Hats, water, and sunscreen on hot days. Groups stay within sight of the teacher.

If the weather does not cooperate, four of the five methods work indoors. Use a gym, a stairwell, or a two-storey atrium and measure a basketball hoop, a banner, or a ceiling fixture. Only Phase 2 needs the sun; replace it with a second mirror trial from a different distance and note the substitution in the packet. Good targets outdoors: a flagpole, a light standard, a goalpost, a backstop, or the school building itself.

Class-Day Timeline

Five class periods. Days 2, 3, and 4 are outdoors. Day 2 is the one that needs sun, so it is the one to move if the forecast is bad.

Day	Where	What happens	Leaves the room with
Day 1	Indoors	Launch the driving question. Build and calibrate the wedge (Phase 0). Write the Isosceles Triangle Theorem justification. Practise all five setups on a known height, such as a doorway.	A finished wedge and a written justification
Day 2	Outdoors	Method 1, the \(45^\circ\) sight, twice with different sighters. Method 2, shadows, with two readings fifteen minutes apart.	Two heights and a timed shadow table
Day 3	Outdoors	Method 3, the mirror, at two different mirror distances. Method 4, the sighting stick, at two different stick distances.	Four more measurements and two heights
Day 4	Outdoors	Method 5, the geometric mean, with the right angle checked using the wedge. Re-measure anything that looked wrong on Days 2 and 3.	The fifth height and any repeat readings
Day 5	Indoors	Comparison table, percent differences, outlier decision, error bar (Phase 6). Three-minute group defenses.	The finished packet, turned in

Running it in three days instead of five: pre-cut the cardboard wedges before Day 1 and assign each group only three of the five methods, chosen so that the class as a whole still covers all five. Pool the class data on Day 3 so every group has five numbers to compare. Phase 6 is the part to protect — a project that stops after collecting the measurements has skipped the mathematics that makes it worth doing.

Rubric

Twenty points across five criteria. The tool, the diagrams, and the computation are checked for correctness; the comparison is checked for honesty and reasoning.

	4 — Exceeds	3 — Meets	2 — Approaching	1 — Beginning
The wedge and its justification	Legs measured twice and recorded to the nearest millimetre; the justification names the Isosceles Triangle Theorem precisely and analyses how a leg-length error would propagate into the angle.	Legs are congruent and the tool produces a usable sight line; the justification correctly derives \(45^\circ\) from congruent legs and the \(180^\circ\) angle sum.	Wedge works but the justification restates “it is a \(45\)–\(45\)–\(90\) triangle” without deriving it, or appeals to a protractor.	Legs are visibly unequal, or no mathematical justification is given at all.
Diagrams and similarity statements	All five diagrams labeled with every measured quantity; each names the similar triangles in correct corresponding order and states which pair of congruent angles establishes the similarity.	All five diagrams drawn and labeled, each with a similarity statement and the proportion that was solved.	Some diagrams missing or unlabeled, or similarity statements written with the letters out of corresponding order.	Sketches without labels; no similarity statements; proportions appear with no figure to justify them.
Measurement and data records	Every method measured at least twice under deliberately different conditions; units recorded beside every number; the person who took each reading is named; nothing was measured once and trusted.	All five methods measured with complete raw data recorded in consistent units, including eye height and stick height.	Data present but incomplete — a missing eye height, unrecorded units, or a single trial where two were required.	Only final heights recorded; the raw measurements they came from cannot be reconstructed.
Computation	All five heights computed correctly with work shown, converted to a common unit, and each cross-checked with a second relationship where one is available.	All five heights computed with the arithmetic shown, rounded to two decimal places, in a common unit.	Computation shown but with a proportion set up incorrectly — commonly piece over whole mixed with piece over piece, or eye height omitted in Method 1.	Answers appear with no supporting arithmetic, or units are mixed inside a single calculation.
Comparison, outlier, and defense	Percent differences computed for all five; the outlier is traced to a specific physical cause with supporting evidence; a systematic error is identified and its direction argued; the error bar is justified from the observed spread.	Percent differences computed; an outlier identified with a plausible physical cause; a final height reported with an error bar and a stated reason; all four written questions answered.	Differences computed but the explanation is generic — “human error” or “we measured wrong” — or the final number is chosen with no stated reason.	No comparison across methods; a single height reported with no acknowledgement that the five disagreed.

On the honesty clause: a group whose five numbers scattered badly and who explains precisely why can score higher on the last criterion than a group with tight agreement and a vague paragraph. Reporting a measurement you did not take is the one thing that cannot be scored at all.

Extensions

Build a second wedge

Cut one with legs in a \(1:2\) ratio instead of \(1:1\). It no longer gives \(45^\circ\), so \(H=d+e\) fails. Work out the correct relationship from similar triangles and measure with it.

Measure the unreachable base

Pick something whose base you cannot walk to — a tree across a creek, a sign beyond a fence. Two mirror positions and a pair of proportions will still get you the height.

Chain two similar triangles

Use one object of known height to calibrate a distance, then use that distance to find a second object. Track how the uncertainty from step one grows in step two.

Pool the class data

Collect every group's height for the same object. Plot the distribution by method. Which method has the tightest spread across the whole class, and is that the same as the most accurate?

Scale it down and check

Run all five methods on something you can measure directly, such as a basketball hoop at \(10\) ft. Compute each method's true error and rank them honestly.

Program the calculations

Write a short script that takes the raw measurements for all five methods and prints the heights, the mean, and the percent differences. A natural bridge to the Computer Science pages.

Teacher Notes

What to prepare in advance

Pre-cut the cardboard into squares before Day 1. Cutting from a large sheet eats most of a period, and the mathematics is in making the two legs equal, not in the cutting.
Walk the site and pick the targets. Assign each group an object rather than letting them choose — groups left to choose gravitate to the tallest thing on campus, which is usually also the one with the least accessible base. A flagpole, a light standard, a goalpost, and a corner of the building make a good set.
Check the sun. Phase 2 needs a shadow long enough to measure, which means mid-morning or mid-afternoon, not noon. Look at the forecast a week out and be ready to swap Days 2 and 3.
Tape the mirror edges yourself before handing them out, and count the mirrors out and back in. One broken mirror on a field is a much longer afternoon than the project is worth.
Have one long tape per group. This is the piece of equipment that makes or breaks the outdoor days. Groups forced to measure \(78\) feet in \(12\)-foot segments accumulate exactly the kind of error that swamps the comparison in Phase 6.

Where groups get stuck

Forgetting eye height in Method 1. Universal. The triangle stops at the eye, and groups hand in a flagpole that is five feet shorter than it is. Worth catching on Day 2 rather than Day 5, because it is a beautiful example of a systematic error — every trial is wrong by the same amount in the same direction.
Mixing units. A stick in centimetres and a shadow in feet. Require the unit to be written next to every number as it is recorded, not added later.
Piece over whole versus piece over piece in Method 4. The same error as the Unit 2 review sheet question 10, now with a physical consequence. Have them label \(AD\), \(DB\), and \(AB\) on the diagram before writing the proportion.
Sliding the mirror instead of moving themselves. It looks like the same thing and it is not. Watch for it on Day 3.
Averaging too early. Groups want to average the two trials within a method and report one number per method. Insist that both trials survive into the data table — the spread within a method is evidence about that method, and Phase 6 needs it.
Blaming “human error.” The single most common sentence in the write-up, and it earns nothing. Push for the specific measurement, the specific direction, and the specific reason.

How this connects to the unit

Each phase is anchored in a different lesson, so the project reads as a tour of the unit rather than an application of its last idea. Phase 0 is the Isosceles Triangle Theorem from Lesson 2, used constructively — students guarantee an angle by controlling two lengths, which is the cleanest argument for why the theorem is worth having. Phases 1 through 3 are the definition of similarity and scale factor from Lesson 3. Phase 4 is the Triangle Proportionality Theorem from Lesson 4, including the piece-over-whole distinction that the lesson warns about. Phase 5 is the geometric mean from Lesson 5, and it doubles as a Unit 1 review because the string lengths can be checked with the Pythagorean Theorem.

Congruence from Lesson 1 is the quiet piece: it appears every time a group repeats a method and expects the same answer, and Phase 6 is where they confront the fact that repeated physical measurements are never congruent. That is worth saying out loud on Day 5.

The assessment connection is direct. Every skill in the lesson map appears in at least one phase, and Phase 6 is the only place in the unit where students defend a number they produced themselves.

Grading load

One packet per group, five criteria, twenty points. The five diagrams and the Phase 6 write-up are where the reading time goes; the wedge and the computations can be checked in a few minutes each with the data table in hand. The three-minute defenses are scored live on the last criterion, which keeps them from becoming a sixth thing to grade later.

Final Deliverables Checklist

Print this page and clip it to the front of the packet. Every box must be checked before the packet is turned in.

Phases 0–3

Cardboard wedge with a working plumb line
Both leg lengths measured, recorded, and compared
Written justification naming the Isosceles Triangle Theorem
Method 1: distance \(d\), eye height \(e\), two sighters, height computed
Method 2: stick height, both shadows, both times of day, height computed
Method 3: \(e\), \(d\), and \(D\) at two mirror positions, heights computed
Labeled diagram and similarity statement for each method so far

Phases 4–6

Method 4: \(DE\), \(AD\), and \(AB\) at two stick positions, heights computed
Method 5: \(AD\), \(DB\), right angle verified with the wedge, height computed
Labeled diagram and similarity statement for Methods 4 and 5
All five heights converted to one common unit
Comparison table complete, with the mean and all percent differences
Outlier identified, with a specific physical cause and a keep-or-discard decision
Final height reported as \(H=\underline{\quad}\pm\underline{\quad}\), with the \(\pm\) justified
All written questions from every phase answered
One decision prepared for the three-minute defense

U2

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