Compare and classify geometric shapes based on their properties and sizes; understand that attributes belonging to a category also belong to all subcategories; classify two-dimensional figures in a hierarchy based on properties
You have sorted shapes by their sides and angles. A quadrilateral is a polygon with four straight sides. A shape belongs to every family whose rules it meets. A more specific family name does not remove a broader one.
Mia's sandpit has four sides of 4 m and four right angles. A rectangle has four right angles and equal opposite sides. A square has four right angles and four equal sides. The sandpit meets both rules, so it is a square and a rectangle. Every square is a rectangle. The reverse is not always true: a rectangle can have unequal neighbouring sides.
Here, quadrilaterals have sides that do not cross. A parallelogram is a quadrilateral with two pairs of parallel opposite sides. Its opposite sides are equal in length. Going round it, the lengths repeat: one length, another length, then those two again. Neighbouring sides need not match. The reverse also holds: if both pairs of opposite sides are equal in length, the quadrilateral is a parallelogram.
This rectangle is one parallelogram. Opposite lengths match: 7 cm with 7 cm, and 3 cm with 3 cm.
Rectangles are parallelograms with four right angles. Rhombuses are parallelograms with four equal sides. Squares have both extras. A parallelogram with no right angles and unequal neighbouring sides fits neither smaller family. Knowing only that opposite sides are parallel does not tell you which extras the shape has.
A hierarchy diagram shows families inside broader families. Put parallelograms inside quadrilaterals. Inside parallelograms, rectangle and rhombus regions overlap. Put squares in their overlap because squares meet both rules. The path below follows just the rectangle branch. Each later family is inside the one before it.
Read this as a family path, not as shapes getting smaller. The rhombus branch is not shown.
To sort a collection, check each shape separately and keep every matching name. Then count only what the question asks for. Mia's square sandpit has five of our family names: quadrilateral, parallelogram, rectangle, rhombus and square. If asked for its most specific name, choose square. That name includes both the equal-side and right-angle rules.
Regular polygons have all sides equal and all angles equal. Squares are regular. Non-square rectangles have unequal neighbouring sides; non-square rhombuses have unequal angles. So a parallelogram need not be regular. To challenge an 'every' claim, find a counterexample: a shape in the named family that lacks the claimed property.
This rectangle is a parallelogram, but its unequal side lengths make it irregular.
Turning a shape leaves its side lengths and angles unchanged. A uniform enlargement makes the whole shape bigger by the same scale factor, keeping every angle unchanged. With scale factor 2, every side doubles. Equal sides stay equal and unequal sides stay unequal; parallel pairs stay parallel. Turning and uniform enlargement preserve shape families and regularity.
A shape belongs to every family whose rules it meets, from the biggest group down to the most exact match.
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A family name tells you which properties a shape must have. To check a claim about every member of a family, compare the rules carefully.
Every square is a rectangle. A square has four right angles, and its opposite sides match because all four sides match. It therefore meets the rectangle rules.
The reverse needs its own check. Consider a rectangle whose neighbouring sides have different lengths. It meets the rectangle rules but lacks the four equal sides a square needs. That one example disproves the claim that every rectangle is a square.
A counterexample must belong to the family the claim starts with. A triangle cannot disprove a claim about every rectangle because it is not a rectangle.
Start a family diagram with quadrilaterals, the four-sided polygons. Draw the parallelogram region inside it. Every parallelogram belongs there because it has four straight sides.
Inside parallelograms, draw a rectangle region and a rhombus region that overlap. The rectangle rule adds four right angles. The rhombus rule adds four equal sides. Squares meet both rules, so put the square family in the overlap.
Do not put all rectangles inside rhombuses. Some rectangles have unequal neighbouring sides. Do not put all rhombuses inside rectangles either. Some rhombuses have no right angles.
The regions need not fill the whole parallelogram region. Parallelograms with unequal neighbouring sides and no right angles belong outside both smaller regions.
Turning a rectangle changes where its sides point on the page. It does not change which sides are parallel or how its corners meet. The rectangle keeps its family names.
A uniform enlargement makes a bigger copy with all angles unchanged. At scale factor 2, every side becomes twice as long. Opposite sides still match, parallel pairs stay parallel, and neighbouring sides that differed before still differ afterwards. A non-square rectangle remains a non-square rectangle.
Regularity is a separate check. A regular polygon needs every side equal and every angle equal. A rectangle with unequal neighbouring sides is irregular even though it belongs to the rectangle and parallelogram families. A square is regular and also belongs to both of those families and the rhombus family.
Use the stated properties to classify a shape. Its size or position on the page does not replace those checks.
Last time you checked whether sides and angles all matched. Now follow the shape families on the playground plan. Mia's sandpit has four sides of 4 m and four right angles, as in the lesson.
Does the sandpit belong to the quadrilateral family?
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