Sort quadrilaterals by the properties they inherit

diagnostic. A convex quadrilateral has four equal sides and four right angles. How many of the names rectangle, rhombus and square apply? Enter the count and justify each name from its properties.

model. Use the exact six-card collection A–F. A rectangle is a quadrilateral with four right angles; a rhombus has four equal sides; a square has both properties. Which cards are rectangles, including squares, and how many are there? Build the rest of the hierarchy to justify your choices.

1. Display the exact side and angle data for A–F and read the inclusive definitions. A quadrilateral has four sides; a parallelogram has two pairs of parallel opposite sides; a rectangle has four right angles; a rhombus has four equal sides; a square meets both the rectangle and rhombus definitions. Treat supplied properties as evidence rather than judging a sketch by eye.
2. Sort by one property at a time. Four right angles place A, B, E and F in the rectangle category. Four equal sides place A, C and F in the rhombus category. The overlap A and F is the square category; a card can satisfy both tests.
3. Draw a hierarchy in words or arrows: quadrilateral contains parallelogram; parallelogram contains rectangle and rhombus; square belongs within both rectangle and rhombus. Every square inherits the properties of both parent categories. The four-right-angle condition ensures a rectangle's opposite sides are parallel; rhombi also have parallel opposite sides.
4. Count distinct cards carefully. There are four rectangles, three rhombi and two squares. In the rectangle-or-rhombus collection there are only five different cards because A and F belong to both groups. Use B to refute 'every rectangle is a square' and C to refute 'every rhombus is a rectangle.'

Worked example: A, B, E and F are rectangles, so the count is 4. A and F are also squares and rhombi; C is a rhombus with no right angles. D is a parallelogram with neither four equal sides nor four right angles. All six are parallelograms and quadrilaterals.

p1. Support: use only cards A–F. How many rectangles are there when squares are included? Enter the count and list their card IDs.

p2. Core: use only cards A–F. How many rhombi are not squares? Enter the count, name the card and cite the property that excludes it from the square category.

p3. Challenge: use only cards A–F. How many different cards are rectangles or rhombi, with a card that is both counted just once? Enter the count, list the IDs and explain how you avoided duplicate counting.

e1. Independent check: use only the same six cards A–F. How many cards belong to exactly one of the rectangle and rhombus categories? A card belonging to both or to neither does not qualify. Enter the count, list the qualifying IDs and justify each inclusion and exclusion using side or angle properties.

transfer. A learner says, 'Every square is a rectangle, so every rectangle is a square.' Which direction is valid? Use the supplied collection to justify your answer.