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.

Answer: 3

Teacher response: All three names apply. Four right angles make it a rectangle, four equal sides make it a rhombus, and satisfying both conditions makes it a square. The categories are inclusive.

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.

Answer: 4

Teacher response: B and E are rectangles with unequal adjacent sides, and A and F are squares that also satisfy the four-right-angle rectangle definition. The IDs are A, B, E and F: two plus two equals four.

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.

Answer: 1

Teacher response: The rhombi are A, C and F. Removing the two squares A and F leaves C. Its four sides are equal, but its 60- and 120-degree angles are not right angles.

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.

Answer: 5

Teacher response: The distinct IDs are A, B, C, E and F. Four rectangle memberships plus three rhombus memberships counts A and F twice; subtract their two duplicate memberships to get 4 + 3 - 2 = 5 cards. D meets neither definition.

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.

Answer: 3

Look for:

• Independent rubric, 1 point: enters 3 and identifies exactly B, C and E.
• Independent rubric, 1 point: explains B and E have four right angles but not four equal sides, while C has four equal sides but not four right angles.
• Independent rubric, 1 point: excludes A and F because they satisfy both definitions and excludes D because it satisfies neither. Accept a correct labeled hierarchy or property checklist as written evidence; the numerical count alone earns at most the first point.

Teacher response: B and E are rectangles with unequal adjacent sides, so they are not rhombi. C is a rhombus with no right angles, so it is not a rectangle. These three qualify. A and F belong to both categories; D belongs to neither. Equivalently, 4 + 3 - 2 × 2 = 3 removes both memberships of each of the two squares.

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.

Answer: Every square is a rectangle because a square has four right angles. The reverse is false: B has four right angles but consecutive side lengths 6, 3, 6, 3, so its sides are not all equal and it is not a square. E is another valid counterexample. A property passing to a subcategory does not make the two categories identical.

teach-1. 5 minutes: Give the diagnostic, requesting all applicable names. Use the support or ready route. Read the exact A–F card data and agree that sketches will not be used to measure or prove properties.

teach-2. 10 minutes: Model the rectangle test, then let partners apply the rhombus test. Place A and F in the overlap and draw the inclusive hierarchy. Ask learners to name one property a square inherits from each parent category.

teach-3. 15 minutes: Support pairs complete p1 and p2 with a checklist for each card. Ready pairs complete p2 and p3, marking duplicate membership before counting distinct cards. Ask each pair to defend a category choice using exact properties and to give B or C as a counterexample to a reversed hierarchy claim.

teach-4. 10 minutes: Allow six minutes for e1 independently, then use the three-point rubric to inspect the selected card IDs and explanations. Discuss the transfer claim. Revisit definitions with learners earning 0–2 points; learners earning 3 redraw the hierarchy with every card placed in all its valid categories.

misconception. A square cannot be a rectangle because a rectangle must have two long sides and two short sides.

Teacher response: Return to the rectangle definition: four right angles. Ask whether any part requires unequal adjacent side lengths. Check A and F against that definition, then compare B. All pass the rectangle test, while only A and F also pass the four-equal-sides test.

support. Give a two-column checklist headed 'four right angles' and 'four equal sides.' Check each supplied card against both definitions before naming it. Use A and B together to show that equal adjacent side lengths are permitted, but not required, for rectangles.

extend. Make a new exact property card for a rectangle that is not a square, and one for a rhombus that is not a rectangle. Explain which hierarchy arrows still apply. Then describe why no card can be a square that fails the rectangle test.

support-route. If the learner accepts only 'square,' put three definition cards beside the same shape. Check each requirement separately: four right angles, four equal sides, then both. Recheck with card F and ask for all three names before sorting the full collection.

ready-route. If the learner names all three categories and cites their properties, ask whether every rectangle is a square. Use B, with side lengths 6 and 3, as a counterexample before constructing the hierarchy.