The quadrilateral family

Classroom resources

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Students will be able to

• I can name a quadrilateral from its properties.
• I can explain why a square is also a rectangle and a rhombus.
• I can use parallel sides to find angles in a parallelogram.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need parallel lines and co-interior angles (Parallel lines unit).

The one idea: a name is a list of properties, and a shape with every property on a list belongs to that family. This is the unit's hinge: students who hold the everyday meaning ("a rectangle is longer than wide") cannot reason about the family (Sam's mistake).

Definitions: this site follows the Australian Curriculum glossary, where a trapezium has at least one pair of parallel sides, so every parallelogram (and rectangle, rhombus and square) is a trapezium too. Some textbooks say exactly one pair; mention it if yours does.

Running the lesson

1. Predict: take a vote; most say no. Keep the vote visible until after Investigate.
2. Run: ask what the board added to its name as the rectangle appeared.
3. Investigate: build a family tree on the board, from quadrilateral down to square.
4. Modify: for the edge case, ask what co-interior angles say about the next corner.
5. Make: ask why a trapezium can't have exactly three right angles.

The hard part (on purpose)

Accepting that one shape can have several names. Ask students to read a square's properties against each name's list, out loud, instead of telling them the answer.

When students get stuck

Answers to the checks

1. square or rectangle?: Is a square a rectangle?
   Yes. A rectangle is a quadrilateral with four right angles, and a square has four right angles. It is a special rectangle, with its sides equal too.
2. a rectangle that is not a square: Drag the corners to make a rectangle that is not a square. What else does the board say it is? Was your prediction right?
   
   A rectangle is also a parallelogram. Make the sides equal as well and it becomes a square, which is also a rectangle.
3. Properties first: Match each quadrilateral to its defining property.
   trapezium → A. at least one pair of parallel sides; parallelogram → D. two pairs of parallel sides; rhombus → B. four equal sides; rectangle → E. four right angles; kite → C. two pairs of equal sides next to each other
   A square has four equal sides and four right angles, so it fits the rhombus, rectangle and parallelogram rows too.
4. Find the mistake: Sam decides whether a square is a rectangle. Click the line where his reasoning goes wrong.
   line 4
   Line 4: having an extra property (equal sides) does not take away the rectangle's properties. A square has four right angles, so it is a rectangle: a special one.
5. Always true?: Which is always true?
   Every rhombus is a parallelogram
   A rhombus has two pairs of parallel sides, so it has the parallelogram's property: every rhombus is a parallelogram.
6. A rhombus: Make a rhombus that is not a square.
   
   Four equal sides, no right angles. It is also a parallelogram and a kite.
7. A plain parallelogram: Make a parallelogram that is not a rectangle and not a rhombus.
   
   For example (1, 1), (7, 1), (9, 5), (3, 5).
8. A kite: Make a kite that is not a rhombus.
   
   Two pairs of equal sides next to each other, but not all four equal.
9. One right angle: Edge case: a quadrilateral has four equal sides and one right angle. What is it?
   A square
   A rhombus is a parallelogram, so angles next to each other add to 180° (co-interior). One right angle makes them all right angles: a square.
10. Most special name: A quadrilateral has two pairs of parallel sides, no right angles, and sides that are not all equal. What is its most special name?
    Parallelogram
    Two pairs of parallel sides, nothing more: a parallelogram.
11. All the names: Which names fit a square?
    Square, rectangle, rhombus, parallelogram, trapezium and kite
    A square has all these properties, so it belongs to all these families.
12. Angles in a parallelogram: One angle of a parallelogram is 70°. What is the angle next to it, in degrees?
    110
    Neighbouring angles of a parallelogram are co-interior: 180 − 70 = 110°.
13. Review: triangles: Review: an isosceles triangle has two angles of 35°. What is the third, in degrees?
    110
    180 − 35 − 35 = 110°.
14. a right-angled trapezium: Make a trapezium that is not a parallelogram, with at least one right angle.
    
    For example (1, 1), (9, 1), (6, 5), (1, 5): one pair of parallel sides and two right angles.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › A rhombus

Open task: any program that does what the step asks.

one change at a time › A plain parallelogram

Open task: any program that does what the step asks.

one change at a time › A kite

Open task: any program that does what the step asks.

one change at a time › One right angle

Open task: any program that does what the step asks.

a right-angled trapezium

Open task: any program that does what the step asks.

Stretch and support

Stretch: Draw a Venn diagram of rectangles and rhombuses. What goes in the overlap? Where do parallelograms go?

Support: Use a property checklist (parallel pairs, equal sides, right angles) for every shape before naming it.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "The quadrilateral family".

The idea: Classifying quadrilaterals and their relationships: I can Students classify quadrilaterals (square, rectangle, rhombus, parallelogram, kite, trapezium) using side and angle properties, and identify hierarchical relationships.

What understanding looks like:
- Correctly classifies any standard quadrilateral using its properties.
- States at least two hierarchical relationships between quadrilateral types.
- Does not rely on orientation to classify (e.g. recognises a rotated square).

Mistakes students often make here:
- A student says a square is not a rectangle (Sam's mistake).
- A student names a rhombus a diamond and stops there.
- A student cannot make a kite on the board.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Classifying triangles by sides and angles), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "The quadrilateral family".

Exit ticket

A shape has two pairs of parallel sides and four equal sides but no right angles. Give all its names.

Rhombus, parallelogram and kite.

What each step is for

• square or rectangle?: Committing to whether a square is a rectangle surfaces the everyday meaning of 'rectangle' (longer than it is wide) against the property definition.
• a rectangle that is not a square: Dragging an irregular quadrilateral into a rectangle shows the properties appearing one by one (parallel arrows, right angles) and the board's 'also' names.
• a family of properties: Each name is a list of properties, and having all of one name's properties puts a shape in that family; Sam lets an extra property exclude a square from the rectangles.
• one change at a time: One change at a time on the board: a rhombus that is not a square, a parallelogram that is neither, a kite, then the edge case of four equal sides with one right angle.
• naming quadrilaterals: Naming from properties, a family question and an angle from parallel sides, mixed; a square excluded from the rectangles is the planted slip; the review keeps isosceles triangles fresh.
• a right-angled trapezium: Building a trapezium with a right angle that is not a parallelogram combines a parallel pair with a right angle while avoiding a second parallel pair.

Where it leads

Next: Polygons (node 3): regular or irregular, convex or concave.

Where it leads: properties of diagonals and congruence proofs for quadrilaterals (Years 8 to 10).