Lesson type: Discovering — noticing that same-angle right-angled triangles share fixed side ratios, whatever their size.
Stepping Stone: recognise that right-angled triangles with the same angle are similar, so the ratio of two chosen sides is constant.
NEO Cornerstones: Connection · Movement · Reflection · Creativity · Rest · Nutrition
45-minute shape: Connect 12 · Move 12 · Reflect 10 · Create 6 · Rest/Nourish 3 · Evidence 2
Choose only the support that helps now. Every route returns you to this lesson.
Right-angled triangle — a triangle with one 90° corner. The angle we watch is marked θ ("theta").
Similar — two shapes are mathematically similar when their corresponding angles are equal and their corresponding sides are in the same ratio. One is then an exact scale-factor enlargement of the other.
Ratio — one length divided by another. Here we compare the vertical side with the horizontal side: vertical ÷ horizontal. (Next lesson we give these sides their proper names.)
What to do: drag the scale-factor slider to enlarge and shrink the triangle. Every side length is multiplied by the same scale factor, and the angle θ stays the same, so the triangle stays similar. Watch the three side lengths change — and watch the ratio vertical ÷ horizontal.
What you are trying to notice: the sides all change as you scale, but the ratio vertical ÷ horizontal locks at one value and never moves. That fixed number belongs to the angle, not the size.
Carry this forward: for a fixed angle, vertical ÷ horizontal is always the same, however big the triangle. The ratio is a property of the angle.
What to do: press Add a bigger one to stack another triangle on top, all sharing the same corner and the same angle θ. Each new triangle is a scaled-up copy. Work out vertical ÷ horizontal for each as it appears.
What you are trying to notice: however tall the stack grows, every triangle in it gives the same ratio. Same shape → same ratio, at any size.
Carry this forward: a whole family of similar triangles shares one ratio. The ratio does not grow when the triangle does.
What to do: each card shows two right-angled triangles with their vertical and horizontal sides given. Work out vertical ÷ horizontal for each and decide whether the two triangles are similar or not similar, then check.
What you are trying to notice: two right-angled triangles are similar exactly when their ratios are equal — not when they are the same size. Equal ratio means the angle θ is equal, which is what makes them similar.
Carry this forward: to tell if two right-angled triangles are similar, compare their ratios — not their sizes. Equal ratios mean equal angles.
What to do: here is a right-angled triangle the same shape as one you know (its ratio vertical ÷ horizontal is given). You are told one side. Use the locked ratio to work out the side you cannot see. Type your answer and check.
What you are trying to notice: once you know the ratio for a shape, one measurement is enough to find the rest. You can reach a length you never measured — the first taste of what trigonometry is for.
Make it your own: on the Scratchpad, choose a ratio, draw a right triangle to fit it, cover one side and challenge a friend to find it.
Watching everything about a triangle change — every side, its whole size — while one number stays perfectly still is a quietly beautiful thing. Take a breath before moving on.
Notice one thing: when the triangle grew and grew but the ratio did not move at all, what did that feel like? There is no right answer — just what your own attention noticed.
You can jot it on the Scratchpad, say it aloud, or simply hold it in mind. You can pause here and return whenever you are ready.
Mode: intellectual. This mathematics does not live in food or the body, so we name honestly what the learning nourishes.
What it feeds: today's idea grows one of the most powerful habits in all of mathematics — looking for what stays the same while everything else changes. The triangle grew, every length shifted, yet one ratio held perfectly still. Trained to hunt for that kind of invariant, a mind starts to find hidden order everywhere: the constant that survives the chaos is often where the real understanding lives. It also feeds a quiet confidence — that careful reasoning can pin down something exact even when the picture keeps moving.
Carry one idea forward: when something feels like it is changing wildly, ask — is anything staying the same? That question is the start of a great deal of mathematics.
Your task: show that the ratio belongs to the angle, not the size.
Submission choices: an annotated Scratchpad image, a written solution, a labelled poster, or a spoken/video walkthrough.
Filename pattern: NEO_MATH_TR_THE_CONSTANT_RATIO_FirstName_Date
| Criterion | Approaching Proficiency | Developing Proficiency | Proficiency | Refining Proficiency |
|---|---|---|---|---|
| Understanding | Knows a bigger triangle can be the same shape. | Knows same-shape triangles share their angles. | Explains that same angle → constant side ratio. | Explains why scaling cannot change the ratio (similarity). |
| Application | Works out vertical ÷ horizontal for one triangle. | Works it out for two and compares. | Uses the ratio to decide if two triangles are similar. | Uses a known ratio to find a hidden side. |
| Communication | Records a ratio with limited labelling. | Labels the vertical and horizontal sides. | Shows the vertical ÷ horizontal working clearly. | Explains a method another learner could follow. |
| Reflection | Names one thing noticed. | Spots that the ratio did not change. | Explains that the ratio belongs to the angle. | Connects the constant ratio to finding unknown lengths. |
Use support in this order: 🤔 Socratic prompt → 💡 hint → ✅ check → 👀 worked solution. The worked solution unlocks separately for each question, once you have used the prompt, used the hint and checked two answers.
Use the Scratchpad to predict, sketch, test and revise.
You can use this board, paper or another way of recording your thinking. The Scratchpad is optional.
Angles are in degrees. Examples: 9 × sin(37) = 5.42; sin⁻¹(0.5) = 30