Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
Official wording from the Common Core State Standards for Mathematics (Β© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Shapes can be sorted into categories, and categories can sit inside one another. A square is a special kind of rectangle, and a rectangle is a special kind of parallelogram. The big idea here is that any property true of every member of a category is automatically true of every member of its subcategories. Since all rectangles have four right angles, and squares are rectangles, all squares have four right angles too.
This kind of reasoning is about definitions and logic, not appearances. Many students think a square cannot be a rectangle because it 'looks different', or that a shape turned on its point stops being a square. Fifth graders learn to check a shape against the defining properties: four sides, pairs of parallel sides, right angles, equal side lengths. If a shape has all the properties a category requires, it belongs, whatever its size or orientation. Being able to say 'a rhombus has two pairs of parallel sides because every rhombus is a parallelogram' is the start of the deductive reasoning used in geometry proofs years later.
Students often think the categories are separate. A square has four right angles and opposite sides parallel and equal, so it meets every requirement of a rectangle.
A square rotated 45 degrees is sometimes called a diamond and judged not to be a square. Rotating a shape does not change its properties.
Knowing that squares are rectangles, students may conclude all rectangles are squares. Properties pass down to subcategories, not up to larger categories.
If every rectangle a student has seen was long and thin, they may doubt a nearly square rectangle. Varied examples help.
Every parallelogram has opposite sides parallel and opposite angles equal. A rhombus is a parallelogram with 4 equal sides. A rhombus has one angle of 70 degrees. What are its other three angles?
Answer: The angles are 70, 110, 70 and 110 degrees.
Give students a set of quadrilateral cards and a list of properties. They check each property for each card, then discuss which properties every card in a group shares. Asking 'is it always, sometimes or never true?' for statements like 'a rectangle is a square' sparks productive debate.
Assessment items ask whether statements about shape categories are true or false, which properties a given shape must have, or why a square has a property listed for rectangles.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: D) Square
A square is a rectangle, so it inherits the four right angles. Rhombuses, trapezoids and parallelograms do not have to.
Answer: A) All squares are rectangles
A square has every property a rectangle needs, so all squares are rectangles. The reverse is not true.
Answer: B) Rhombus
A rhombus is a parallelogram, so it has two pairs of parallel sides.
Answer: 115 (also accepted: 115 degrees)
The opposite angle is also 65 degrees. The other two share 360 - 130 = 230 degrees equally, so each is 115 degrees.
Answer: D) Yes, turning does not change its sides or angles
Its four sides are still equal and its angles are still right angles, so it is still a square (and also still a rhombus).
Answer: 22 (also accepted: 22 cm)
The rectangle inherits equal opposite sides, so the sides are 8, 3, 8 and 3 cm. 8 + 3 + 8 + 3 = 22 cm.
A full lesson with slides, activities and an exit ticket on attributes of shape categories, pitched to grade 5 and editable in PowerPoint or Google Slides.
Make a lesson βA printable, differentiated worksheet on 5.G.B.3 with an answer key, ready in about a minute.
Make a worksheet βTurn attributes of shape categories into a quiz students answer online that marks itself, with a class summary for you.
Build a test βYes. A rhombus is a quadrilateral with four equal sides, and a square has four equal sides, so every square is a rhombus as well as a rectangle.
It teaches students to reason from definitions, so they can work out properties of shapes they have not studied directly. This logic is the basis of later geometry.