Lesson type: Core theorem-discovery lesson
Stepping Stones: identify the square-area relationship on a right-angled triangle; state the relationship in words; use the capital/lower-case side-label convention; build a² + b² = c² when c is the hypotenuse length.
NEO Cornerstones: Connection · Movement · Reflection · Creativity · Rest · Nutrition
45-minute shape: Connect 15 · Move 15 · Reflect/Create 7 · Rest/Nourish 3 · Evidence 5
Choose the support that is useful today. The Scratchpad stays inside this lesson.
What to do: drag point B or C to change the right-angled triangle. You can also drag the whole construction to reposition it. Try at least four different triangles.
What you are trying to notice: compare the areas of the two squares on the shorter sides with the area of the square on the hypotenuse. Ask whether the same relationship survives every change.
Drag the blue or green point to change the right-angled triangle. Drag anywhere else on the completed construction to move the whole triangle-and-squares shape around the workspace. Watch what happens to the areas of the three squares.
First, compare the three areas without revealing a calculation.
Try at least four different triangles. Does the same area relationship survive?
Complete the statement in your own words before any formal theorem is named.
You have just investigated an area relationship. Now move through the representations one layer at a time.
What to do: use the five stage buttons in order. At each stage, pause and connect the new label or symbol to the triangle before moving on.
What you are trying to notice: the formula is not a separate rule. It is a compressed statement about the areas of the three constructed squares.
Use the buttons in order. Each stage reveals one new layer of the mathematics. Do not rush to the formula: make sure each symbol still has a geometric meaning.
Angle C is marked as a right angle. Which side lies directly opposite vertex C?
Your task: use the diagram below as your evidence. Work through the six questions in order. You are not being asked to invent a proof from memory — you are being asked to read the geometry carefully and explain what each part tells you.
You can use this sentence frame, change it, or explain the same reasoning in your own way:
“The right angle is at vertex ___. The side opposite it is ___, so this is the hypotenuse. The lower-case letter ___ names the length of that side. A square built on a side of length ___ has area ___. The other two squares have areas ___ and ___. The diagram shows that ___ + ___ = ___, so in this labelled right triangle the relationship is a² + b² = c².”
What you are trying to notice: the formula is not true because the letter c is magically special. In this convention, c is used for the hypotenuse length because it is opposite vertex C, where the right angle is located in this diagram.
What to do: create a small teaching card for someone who has only memorised a² + b² = c² and does not yet know what it means.
Choose one format: annotated diagram, three-frame comic, short voice-script, metaphor, or a colour-coded Scratchpad page.
What you are trying to notice: a strong explanation can begin with the geometry and arrive at the symbols, rather than beginning with a memorised formula.
You do not need to hold the whole diagram and formula in attention at once.
Pause after any step. The mathematical relationship will still be there when you return.
This lesson uses intellectual nourishment. The mathematics is strengthening ways of noticing and reasoning, rather than using a food or health context.
Which mathematical habit have you been feeding today?
Carry one idea forward: complete the sentence, “The part of Pythagoras worth keeping is…”
Create one annotated theorem card that begins with a right-angled triangle and three squares, and ends with the symbolic statement of the theorem.
Your evidence must show:
Submission choices: annotated Scratchpad image, one-page poster, short written explanation, or voice/video walkthrough with an accompanying labelled diagram.
Filename pattern: NEO_MATH_PYTH_THEOREM_FirstName_Date
| Criterion | Approaching Proficiency | Developing Proficiency | Proficiency | Refining Proficiency |
|---|---|---|---|---|
| Understanding | Names some features. | Describes the area relationship with some support. | Explains the theorem as a square-area relationship. | Connects geometry and notation precisely and explains the condition on c. |
| Application | Marks a right triangle with support. | Identifies the hypotenuse in a familiar orientation. | Uses the opposite-side labelling convention correctly. | Transfers the convention accurately to a differently oriented right triangle. |
| Communication | Shows an idea with limited explanation. | Uses some correct labels and words. | Communicates a clear geometry-to-formula chain. | Creates a precise explanation that could teach another learner. |
| Reflection | Names one thing noticed. | Identifies one changed idea. | Explains what helped the theorem make more sense. | Identifies a nourishing mathematical habit and a precise next connection. |
Use support in this order: 🤔 prompt → 💡 hint → ✅ check → 👀 worked solution.
A space 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