045i - Problem Solving with Shape Classification

Lesson 45 — Problem Solving and Consolidation: Classifying Shapes

Strand: Space | Descriptor: AC9M7SP02 | Duration: 45 minutes

Learning Intentions

• To consolidate the classification of triangles, quadrilaterals and polygons.
• To identify a shape from a minimal set of clues and justify the identification.

Success Criteria

I can:

1. Name a shape from a list of properties.
2. Decide whether the clues given are enough to determine a shape uniquely.
3. Justify a classification by referring to definitions.
4. Construct my own set of clues for a chosen shape.

Warmup

(6 minutes — odd one out, pairs)

For each set, name the odd one out. There may be more than one defensible answer — the justification matters more than the choice.

1. Square, rhombus, rectangle, kite
2. Equilateral, isosceles, scalene, right-angled
3. Pentagon, hexagon, octagon, quadrilateral
4. Parallelogram, rhombus, trapezium, rectangle

Sample answers: 1. Kite — the only one without two pairs of parallel sides. 2. Right-angled — the only classification by angle rather than by sides. 3. Quadrilateral — the only one whose name does not use a Greek number prefix. 4. Trapezium — the only one that is not a parallelogram.

Activities

Activity 1 — Shape Detective (14 min)

Pairs. For each clue set, name every shape that fits, then state whether the shape is uniquely determined.

Clue Set A. Four sides; two pairs of parallel sides; all sides equal; one angle is .

Clue Set B. Three sides; two equal angles; one angle greater than .

Clue Set C. Four sides; exactly one pair of parallel sides; one line of symmetry.

Clue Set D. Four sides; diagonals cross at ; exactly one line of symmetry.

Clue Set E. All sides equal; all angles equal; five sides.

Clue Set F. Four sides; opposite angles equal; no line of symmetry.

Socratic scaffolding for Clue Set A:

Prompt	Purpose
Take the clues one at a time. What does “two pairs of parallel sides” give?	A parallelogram.
Add “all sides equal”.	A rhombus.
Now add “one angle is “.	In a parallelogram, adjacent angles sum to , so if one is they all are.
So what is the shape?	A rhombus with four right angles — a square.
Is it uniquely determined?	Yes, up to size. Any shape fitting all four clues is a square.
Could you have used fewer clues?	”All sides equal” plus “one angle ” plus “parallelogram” is already enough.

Answers:

• A — square. Uniquely determined.
• B — obtuse isosceles triangle. Not uniquely determined: the obtuse angle could be any value between and , e.g. or .
• C — isosceles trapezium. Determined as a type, not a specific shape.
• D — kite. (A rhombus has two lines of symmetry, so “exactly one” excludes it.)
• E — regular pentagon. Uniquely determined up to size.
• F — a general parallelogram (not a rectangle or rhombus, since those have symmetry lines).

Activity 2 — Build Your Own Clues (10 min)

Pairs, then swap.

1. Choose a shape secretly from the full list studied so far.
2. Write exactly three clues that determine it uniquely.
3. Swap with another pair and identify each other’s shape.
4. If a partner’s clues admit more than one shape, work together to find the minimum extra clue needed.

Worked demonstration — for a rectangle:

• Four sides.
• Two pairs of parallel sides.
• Diagonals are equal but adjacent sides are not.

(A square would fail the last clue, so this pins down a non-square rectangle.)

Teacher prompt: which shapes are hardest to pin down in three clues, and why? (The square is easy — it has many distinguishing properties. A general parallelogram is harder, because it is defined largely by what it lacks.)*

Activity 3 — Inquiry: Shapes on a Grid (10 min)

Pairs, square dot paper.

Using only the dots of a grid as vertices:

1. Draw a square, a rectangle that is not a square, and a rhombus that is not a square.
2. Draw a parallelogram with no right angles.
3. Draw an isosceles triangle and a scalene triangle.
4. Can you draw an equilateral triangle with all three vertices on grid dots? Investigate.

Socratic scaffolding for Q4:

Prompt	Purpose
Try it. What happens?	Every attempt is close but slightly off.
How would you check whether a triangle is equilateral?	Measure all three sides carefully.
What lengths are possible between grid dots?	Horizontal and vertical whole numbers, plus diagonals like and .
For an equilateral triangle, what would you need?	Three equal lengths, with the apex the same distance from both base ends.
Take a base of units. How high would the apex need to be?	Roughly units — which is , not a whole number.
Does that land on a dot?	No. And this happens for every base length you try.
Looking back	It is impossible — though proving it fully is beyond Year 7. Noticing why it keeps failing is the valuable part.

Answers: 1–3. Various correct constructions. 4. It is impossible to draw an exactly equilateral triangle with all vertices on square grid points. Students should conclude “impossible, because the height is never a whole number” — a legitimate Year 7 justification.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Name the shape: four sides, two pairs of parallel sides, all angles , adjacent sides unequal.
2. Name the shape: three sides, all angles .
3. Name the shape: four sides, two pairs of adjacent equal sides, exactly one line of symmetry.
4. A shape has five equal sides and five equal angles. Name it and state its number of diagonals.
5. Reasoning. A student says “four equal sides is enough to identify a square.” Give a counterexample and state what extra clue is needed.

Answers: 1. Rectangle; 2. Equilateral triangle; 3. Kite; 4. Regular pentagon, with diagonals; 5. A rhombus with a angle has four equal sides but is not a square. Adding “one angle is ” (or “diagonals are equal”) completes the identification.

Common Misconceptions

Misconception	How to pre-empt it
Assuming any set of clues determines a unique shape.	Clue Set B admits infinitely many triangles. Always ask “is this enough?”
Naming a shape from its appearance rather than its stated properties.	Work from clue lists with no diagram, as in Activity 1.
Forgetting that families nest, so several names may be correct.	A square is legitimately a rectangle, rhombus and parallelogram. Ask for the most specific name.
Confusing “exactly one line of symmetry” with “at least one”.	Clue Set D turns on this. Read the wording precisely.
Believing size matters to classification.	A shape’s name depends on properties, not measurements.
Assuming a construction is possible just because it can be drawn approximately.	The grid equilateral triangle investigation confronts this.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A quadrilateral has four equal sides and diagonals of equal length. What is it?

Answer

Four equal sides make it a rhombus; equal diagonals make it a rectangle. A shape that is both is a square.

E2 (AMC Junior style). How many squares of any size can be found on a grid of squares?

Answer

(Sixteen , nine , four and one .)

E3 (Challenge). A regular polygon has diagonals. How many sides has it, and how many lines of symmetry?

Answer

Testing, gives ✓. A decagon, with lines of symmetry.

E4 (Challenge). In how many ways can a regular hexagon be cut into two congruent pieces by a single straight line?

Answer

Any line through the centre of a regular hexagon divides it into two congruent halves — infinitely many. But lines that are also lines of symmetry number exactly : three through opposite vertices, three through opposite edge midpoints. Discuss both readings.

E5 (Reasoning challenge). A quadrilateral has exactly two lines of symmetry. Name every shape it could be, and explain why a kite is not among them.

Answer

A rectangle (not square) or a rhombus (not square) — the square has four. A kite has exactly one line of symmetry, along its long diagonal only.

Homework

1. Name the shape from each set of clues: (a) Four sides, all equal, all angles . (b) Three sides, two equal, one angle . (c) Four sides, exactly one pair of parallel sides. (d) Six equal sides, six equal angles. (e) Four sides, opposite sides parallel and equal, no right angles, no lines of symmetry.
2. For each, state whether the clues determine a unique shape, and explain: (a) Three sides, all equal. (b) Four sides, two pairs of parallel sides. (c) Four sides, all equal, one angle .
3. Write three clues that uniquely identify (a) a rhombus that is not a square (b) a regular octagon (c) an isosceles right-angled triangle.
4. How many diagonals does a regular polygon with sides have?
5. How many lines of symmetry does each have: (a) equilateral triangle (b) kite (c) rectangle (d) regular pentagon?
6. Reasoning. Explain why “all angles ” is enough to identify a rectangle, but not enough to identify a square.
7. Reasoning. A shape is described as “a quadrilateral with perpendicular diagonals”. Name three different shapes fitting this description.
8. Challenge. On square dot paper, draw a square whose sides are not horizontal or vertical. State its side length using a square root.
9. Challenge. A regular polygon has diagonals. How many sides has it?

Answers: Q1 — (a) square (b) right-angled isosceles triangle (c) trapezium (d) regular hexagon (e) parallelogram. Q2 — (a) yes, an equilateral triangle, up to size (b) no — parallelogram, rectangle, rhombus and square all fit (c) yes, a square. Q4 — . Q5 — (a) (b) (c) (d) . Q6 — four right angles force a rectangle, but the sides may be unequal; a square additionally needs all four sides equal. Q7 — rhombus, square, kite. Q8 — e.g. a square with vertices , with side . Q9 — , so , a dodecagon.