Free / Grades 9-10, with fraction support
Triangle Studio
What stays the same when a right triangle turns or grows, and what evidence lets us predict a length?
3 flexible sessions / about 130 minutes including practice / right-triangle roles, scale and evidence reports
Suggested rhythm: three sessions across one flexible week, on Monday, Wednesday and Friday. Keep the first drawing and explanation for the final report. Take longer when useful.
Reading, watching and paper activities are free. Parents and instructors can save a collection; scheduling it for a student requires Plus or Lifetime. No upload is required.
Before you begin
Recognize a right angle, name a side by its two endpoints, simplify fractions and multiply by a scale factor. Review the linked lessons before the journals.
Tabletop paper only. No cutting, climbing, construction or outdoor surveying. Use paper drawings or the supplied exact model lengths. Do not measure the printed teaching diagrams: they are schematic. Actual ruler readings are approximate and must stay separate from exact model values.
Materials
- 3 sheets per learner: Session journals
Copy the headings into a notebook.
- 2 sheets per learner: Construction and report paper
Reuse one construction sheet across sessions 1 and 2; keep a separate final report or dictate both.
- 1 per learner: Pencil
Point, speak or use an accessible writing tool.
- 1 shared, optional: Metric ruler and calculator
Use supplied model lengths and written fraction arithmetic.
- 1 shared: Browser or printed pages
Read transcripts together. Separate worksheet printing is optional.
Choose the support that fits
- More support: point to the named angle first, then trace each side by its two endpoints. State the numerator and denominator before calculating.
- More challenge: compare two shapes that have only one matching length. Explain why this alone cannot prove similarity.
- Access options: use every supplied exact length instead of measuring. Dictate a private report; color, fine motor drawing, upload and public presentation are not required.
What good evidence looks like
- Name the right angle and the chosen acute angle.
- Identify sides by endpoints, not page position.
- Match a ratio to the needed pair of sides.
- Check corresponding angles or a common scale before transferring a ratio.
- Distinguish exact model facts from measurements and unsupported guesses.
These are discussion criteria, not a new automatic score. Existing lesson and worksheet records keep their own subjects. Checking off a planned task does not demonstrate mastery or add a second grade.
Session 1 / about 40 minutes
Turn the page, keep the roles
Goal: Identify three side roles relative to a named acute angle and separate a rotation from choosing a different angle.
Preparation / about 8 minutes of adult support: Use journal 1 and a construction sheet. Keep a first explanation on that sheet. Drawing and measuring are optional; the exact model data are enough.
In the separate worked diagram, E is the right angle and D is our chosen angle. DF is across from E, so it is the hypotenuse. EF is across from D; DE touches D but is not the hypotenuse. Turn a drawing without changing its labels. A side does not get a new job just because it is now at the top of the page. Choose F instead, however, and the two legs exchange roles. The hypotenuse keeps its job. It has not applied for a transfer.
Before naming a side, ask which angle we mean. What changed when the page turned? What changed when we chose another angle?
- Name the sides of a right triangle / about 10 minutes
- Name Right-Triangle Sides / about 10 minutes
- Turn and label my triangle / about 20 minutes
Fun fact: A quarter-turn moves every point, but it does not change which endpoints a side connects. Page direction is not a side name.
S3U / Triangle Studio / Journal 1 of 3
One drawing, two chosen angles
Exact model ABC is right at B: AB = 3 cm, BC = 4 cm, AC = 5 cm. Optionally draw B at a paper corner, A 3 cm up one straight edge and C 4 cm along the perpendicular edge, then join A to C. Use the model values even if your drawing is imperfect. Turn your sheet a quarter-turn without relabeling it.
| Angle or observation | Opposite / adjacent leg / hypotenuse |
|---|---|
| Choose A before turning | |
| Choose A after turning | |
| Choose C instead | |
| Optional measured AC (nearest 0.1 cm) |
Keep a first explanation on the construction sheet: does turning a triangle change its side roles? Explain which angle you chose.
If you measured AC, label it approximate. Explain why a ruler reading is not proof that the model hypotenuse is exactly 5 cm. If not, write "used supplied model".
Session 2 / about 45 minutes
Grow every side, not just one
Goal: Compare corresponding ratios after uniform scaling, and detect a stretch that does not preserve them.
Preparation / about 9 minutes of adult support: Use journal 2 and the retained construction sheet. Add a second model or describe its supplied lengths. Keep all ratio comparisons tied to named angles.
The worked models UVW and RST have corresponding angles U and R. Their lengths are 5, 12, 13 and 10, 24, 26. Every side doubles; the ratio 12/13 becomes 24/26, which simplifies back to 12/13. For the selected angle, sine uses opposite/hypotenuse, cosine adjacent/hypotenuse, and tangent opposite/adjacent. A bigger triangle need not have a bigger ratio. Stretch only one leg and this evidence no longer works. A drawing can look convincing while its fractions politely disagree.
Which sides correspond? What evidence supports a common scale? Why should we not swap the chosen angle halfway through a comparison?
- Choose a right-triangle ratio / about 10 minutes
- Choose Sine, Cosine or Tangent / about 10 minutes
- Compare a scale and a stretch / about 25 minutes
Fun fact: In the worked model, both 12/13 and 24/26 describe the same comparison even though every length in the second triangle is twice as large.
S3U / Triangle Studio / Journal 2 of 3
A scale is not a one-leg stretch
Use ABC from journal 1. Exact model DEF is right at E with DE = 6, EF = 8, DF = 10 cm; D corresponds to A. Separately, GHI is right at H with GH = 6 and HI = 4 cm; choose G. These are supplied model values, not measurements from the pictures.
| Comparison | My fractions and evidence |
|---|---|
| ABC to DEF: scale for each side | |
| From D: sine, cosine and tangent | |
| From F: sine, cosine and tangent | |
| Compare tangent from A, D and G |
Explain why all three matching scale factors matter. On your construction sheet, add DEF or describe how each ABC length changed.
Does GHI preserve the chosen angle from ABC? Support your answer using the two leg ratios, without guessing the missing hypotenuse.
Session 3 / about 45 minutes
Make a claim the evidence supports
Goal: Transfer a justified ratio to a fresh similar triangle, reject a wrong-side calculation and state what cannot be inferred.
Preparation / about 10 minutes of adult support: Use journal 3 and a final-report sheet. Bring the first explanation. A spoken report is welcome; retain a brief note of the evidence and any help used.
A length prediction needs more than a familiar-looking picture. Name the right angle and the selected acute angle, mark the two relevant sides, and explain why the triangles are similar. Only then transfer the ratio. For a separate example, an opposite/hypotenuse ratio of 5/13 and a corresponding hypotenuse of 39 give opposite length 39 x 5/13 = 15. Using adjacent/hypotenuse would answer a different question. When the matching-angle evidence is missing, say what else you need. A question mark can be more accurate than a confident invented number.
Ask the learner to point to the requested side before calculating. What supports the transfer? Which statement would stop being justified if similarity were removed?
- Explain Ratios and Find a Length / about 10 minutes
- Check my triangle prediction / about 15 minutes
- Create my triangle evidence report / about 20 minutes
Fun fact: In the worked example, a correct multiplication can still answer the wrong question if it uses the ratio for a different side. Naming the target comes first.
S3U / Triangle Studio / Journal 3 of 3
My prediction and its evidence
Exact JKL is right at K: JK = 20, KL = 21, JL = 29 cm. Choose L. MNP is similar to JKL, right at N, with P corresponding to L and hypotenuse MP = 58 cm. A draft claims the side opposite P is 58 x 21/29 = 42 cm. A different right triangle has hypotenuse 58 cm but no other information.
| Report check | My calculation and reason |
|---|---|
| From L: three side roles and ratios | |
| MNP: scale and both missing legs | |
| Draft: which side is actually 42? | |
| Different triangle: what is unknown? |
On the final-report sheet, draw or describe the labels, show your ratio and explain what supports transferring it. Do not measure the schematic pictures.
Keep the first explanation and say what you confirmed or changed. Include why hypotenuse 58 alone does not determine both legs, and name any help you used.
What changed in your explanation?
Show your evidence, explain one revision, and choose a question to investigate next. You can keep everything on paper.
Optional: record actual offline learning in Learning Records. Keep reported minutes separate from website time. For an assigned pack, use your existing daily tasks; this page does not award additional completion credit.