Geometric Attributes and Spatial Reasoning

Defining vs. Non-Defining Attributes

When we look at shapes, we can notice many things about them. We might see their color, their size, or which way they are turned. We might also notice how many sides they have or whether they have corners.

In math, some shape attributes are defining attributes. These are the special parts that make a shape the shape it is. Other attributes are non-defining attributes. These are things that can change without changing the shape's name.

This lesson will help you learn how to tell the difference.

What is an attribute?

An attribute is a feature or part of something. Shapes have attributes too.

• A shape may have sides.
• A shape may have corners.
• A shape may be large or small.
• A shape may be blue, red, or green.

Not all of these attributes are equally important in math.

Defining attributes

Defining attributes are the features that tell us what a shape really is. If a defining attribute changes, the shape may become a different shape.

For 2D shapes, defining attributes often include:

• the number of sides
• the number of corners
• whether sides are straight
• whether some sides are the same length
• whether the shape has certain kinds of angles

For example:

• A triangle has 3 sides and 3 corners.
• A rectangle has 4 sides and 4 square corners.
• A hexagon has 6 sides.

These are rules for the shape. That is why they are called defining attributes.

Non-defining attributes

Non-defining attributes are features that do not decide the shape's name. These things can change, but the shape stays the same kind of shape.

Non-defining attributes often include:

• color
• size
• direction or orientation (which way the shape is turned)
• where the shape is placed on the page

For example, a triangle can be:

• small or large
• red or yellow
• pointing up, down, or sideways

It is still a triangle as long as it keeps its defining attributes: 3 sides and 3 corners.

A shape does not change its name just because it turns

Sometimes students think a shape is different when it is turned. But turning a shape does not change the kind of shape it is.

If you turn a square, it is still a square. If you turn a triangle, it is still a triangle.

The defining attributes stay the same even when the shape looks different on the page.

Think: "What must be true?"

To decide whether an attribute is defining, ask this question:

Does this feature have to be true for the shape to have this name?

If the answer is yes, it is probably a defining attribute.

If the answer is no, it is probably a non-defining attribute.

Look at these examples:

• A triangle must have 3 sides. That is defining.
• A triangle does not have to be green. That is non-defining.
• A rectangle must have 4 square corners. That is defining.
• A rectangle does not have to be big. That is non-defining.

Worked Example 1: Finding the defining attribute

A shape is blue. It has 3 sides. Which attribute tells us it is a triangle?

Step 1: Look at the attributes.

• blue
• 3 sides

Step 2: Ask which one decides the shape's name.

Color does not decide the name. A triangle can be any color.

But a triangle must have 3 sides.

Answer: 3 sides is the defining attribute. Blue is a non-defining attribute.

Worked Example 2: Same shape, different look

Look at two shapes:

• Shape A is a small square.
• Shape B is a large square turned like a diamond.

Are they both squares?

Step 1: Ignore size and direction. Those may be non-defining.

Step 2: Check the defining attributes of a square.

• 4 equal sides
• 4 square corners

If both shapes have these attributes, then both are squares.

Answer: Yes. They are both squares. The size and the way the shape is turned do not change its name.

Worked Example 3: Sorting attributes

A rectangle has these attributes:

• 4 sides
• red
• 4 square corners
• large

Which are defining attributes, and which are non-defining?

Step 1: Think about what a rectangle must have.

• It must have 4 sides.
• It must have 4 square corners.

Step 2: Think about what can change.

• It can be red, blue, or any color.
• It can be large or small.

Answer:

• Defining: 4 sides, 4 square corners
• Non-defining: red, large

Worked Example 4: Is it still the same polygon?

A shape has 6 sides. One is small, one is medium, and one is large, but all have 6 sides. Are they all hexagons?

Step 1: Remember the defining attribute of a hexagon.

A hexagon has 6 sides.

Step 2: Check the other feature.

The shapes are different sizes. Size is a non-defining attribute.

Answer: Yes. They are all hexagons because they all have 6 sides.

How this helps you classify shapes

To classify shapes means to sort them into groups. In math, we group shapes by defining attributes, not by non-defining ones.

For example, if you are sorting shapes, you should group together:

• all shapes with 3 sides as triangles
• all shapes with 4 sides and 4 square corners as rectangles
• all shapes with 6 sides as hexagons

You should not sort them only by:

• color
• size
• whether they are turned

Helpful reminders

• Count sides and corners.
• Look for angle clues, like square corners.
• Ignore color.
• Ignore size.
• Ignore which way the shape points.

Quick check

Decide whether each attribute is defining or non-defining.

1. A triangle is yellow.
   Yellow is non-defining.
2. A shape has 4 square corners.
   4 square corners is defining for rectangles and squares.
3. A hexagon is tiny.
   Tiny is non-defining.
4. A polygon has 5 sides.
   5 sides is defining for a pentagon.

Summary

Shapes have many attributes, but only some are used to name and classify them. Defining attributes are the important math rules, like number of sides, corners, and certain angles. Non-defining attributes are things like color, size, and direction.

When you study a shape, ask yourself: What must be true about this shape? That will help you find the defining attributes and choose the correct shape name.

Angle Identification and Right Angles

Angles are all around us. You can see them in the corner of a book, a window, a door, or a picture frame.

In this lesson, you will learn what an angle is, where to find it, and how to tell if an angle is a right angle.

What is an angle?

An angle is made when two lines or sides meet at one point.

The point where they meet is called a vertex.

So, when you look for an angle, look for:

• two sides or lines
• meeting at one point
• to make a corner

That corner is the angle.

What is a right angle?

A right angle is a special angle. It makes a perfect square corner.

A right angle measures:

$$90^\circ$$

You do not need to measure every angle with a tool to notice a right angle. In 3rd Grade, a good way to spot one is to ask: Does it look like the corner of a square or rectangle?

If the answer is yes, it is probably a right angle.

Where can we find right angles?

You can find right angles in many everyday objects:

• the corner of a sheet of paper
• the corner of a book
• the corner of a door
• the corner of a window
• the corner of a tile

Squares and rectangles have right angles at their corners.

How to identify an angle

To identify an angle, follow these steps:

1. Find where two sides or lines meet.
2. Look at the corner they make.
3. Name that corner as an angle.
4. Decide if it is a right angle by checking if it looks like a square corner.

Important idea: Not every place where lines cross makes just one angle. Sometimes lines can make more than one angle. But each angle is still found at the point where the lines meet.

Comparing angles

Some angles are smaller than a right angle. Some are larger than a right angle.

• If an angle looks like a square corner, it is a right angle.
• If it is smaller than a square corner, it is not a right angle.
• If it is wider than a square corner, it is not a right angle.

For this lesson, the most important job is to find the angle and tell whether it is a right angle or not.

Worked Example 1

A notebook has a corner where the top side and side edge meet. Is that corner an angle? Is it a right angle?

Step 1: Do two sides meet? Yes.

Step 2: Do they meet at one point? Yes.

Step 3: Does the corner look like the corner of a square or rectangle? Yes.

Answer: It is an angle, and it is a right angle.

Worked Example 2

Look at the tip of a slice of pizza. Two straight sides meet at one point. Is it an angle? Is it a right angle?

Step 1: Do two sides meet? Yes.

Step 2: Do they form a corner? Yes.

Step 3: Does it look like a square corner? No. It looks smaller.

Answer: It is an angle, but it is not a right angle.

Worked Example 3

A rectangle has 4 corners. How many right angles does it have?

A rectangle is made with square corners.

Each corner is a right angle.

So the number of right angles is:

$$4$$

Answer: A rectangle has 4 right angles.

Worked Example 4

A shape has 3 corners. One corner looks like a square corner. The other 2 do not. How many right angles does the shape have?

Step 1: Count only the corners that look like square corners.

Step 2: There is 1 such corner.

Answer: The shape has 1 right angle.

Tips for spotting right angles

• Think of the corner of a square.
• Think of the corner of a piece of paper.
• Look for corners that are not too narrow and not too wide.
• Check if the sides meet like an "L" shape.

Things to remember

• An angle is made when two lines or sides meet.
• The meeting point is called the vertex.
• A right angle is a square corner.
• A right angle measures \(90^\circ\).
• Squares and rectangles have right angles.

Quick Practice

1. Is the corner of a book a right angle?
2. Is the point of a triangle always a right angle?
3. How many right angles does a square have?
4. If two sides meet but do not make a square corner, is it still an angle?

Quick Practice Answers

1. Yes.
2. No, not always.
3. 4.
4. Yes, it is still an angle, but not a right angle.

Summary

An angle is a corner made when two lines or sides meet at a vertex.

A right angle is a special angle that looks like the corner of a square or rectangle. It measures \(90^\circ\).

When you look at shapes or objects, find the corners, then decide which ones are right angles.

Parallel and Perpendicular Lines

In geometry, lines can relate to each other in different ways. Two important kinds of line pairs are called parallel lines and perpendicular lines.

Learning to spot these lines helps us describe shapes, especially polygons like squares, rectangles, and other flat shapes.

Parallel lines are lines that stay the same distance apart and never meet, even if they keep going forever.

You can think of parallel lines like railroad tracks or the two long sides of a rectangle.

Perpendicular lines are lines that cross and make a right angle. A right angle is a square corner. It measures \(90^\circ\).

You can think of perpendicular lines like the corner of a book, a window, or the place where a wall meets the floor.

How to identify parallel lines:

• Look to see if the lines go in the same direction.
• Check if the space between them stays even.
• Ask: if these lines kept going, would they ever touch? If the answer is no, they are parallel.

How to identify perpendicular lines:

• Look to see if the lines cross.
• Check whether they make a square corner.
• If they make a right angle, then they are perpendicular.

It is important to know that not all crossing lines are perpendicular. Lines can cross without making a right angle.

Also, parallel lines do not cross. If two lines meet, they are not parallel.

Let’s look at these ideas in shapes.

Parallel and perpendicular lines in polygons

A polygon is a flat shape with straight sides. Some polygons have parallel sides, some have perpendicular sides, and some have both.

• A square has 2 pairs of parallel sides. It also has sides that meet to make right angles, so it has perpendicular sides too.
• A rectangle also has 2 pairs of parallel sides and 4 right angles, so it has perpendicular sides too.
• A triangle may or may not have perpendicular sides. A right triangle has one right angle, so it has perpendicular sides. Most triangles do not have parallel sides.
• A trapezoid can have 1 pair of parallel sides.

Helpful clues in common shapes

• Opposite sides of a square are parallel.
• Opposite sides of a rectangle are parallel.
• Any corner of a square or rectangle shows perpendicular sides.
• If a shape has a square corner, the two sides making that corner are perpendicular.

Worked Example 1: Finding parallel sides in a rectangle

A rectangle has a top side, bottom side, left side, and right side. Which sides are parallel?

Step 1: Look for sides that go in the same direction and never meet.

Step 2: The top and bottom sides both go across. They stay the same distance apart, so they are parallel.

Step 3: The left and right sides both go up and down. They also stay the same distance apart, so they are parallel.

Answer: The top and bottom sides are parallel, and the left and right sides are parallel.

Worked Example 2: Finding perpendicular sides in a square

A square has 4 corners. Are the side lines perpendicular?

Step 1: Look at one corner of the square.

Step 2: The two sides meet to make a square corner.

Step 3: A square corner is a right angle, and a right angle is \(90^\circ\).

Answer: Yes. Any two sides that meet at a corner of the square are perpendicular.

Worked Example 3: Looking at a triangle

A triangle has one right angle. Does it have perpendicular sides?

Step 1: A right angle means the corner is \(90^\circ\).

Step 2: The two sides that meet to make that right angle are perpendicular.

Step 3: Check for parallel sides. In this triangle, no sides stay the same distance apart forever.

Answer: The triangle has one pair of perpendicular sides, but no parallel sides.

Worked Example 4: A trapezoid

A trapezoid has one top side and one bottom side that never meet. The other two sides slant. Which sides are parallel?

Step 1: Look for the pair that stays the same distance apart.

Step 2: The top and bottom sides go in the same direction and never meet.

Step 3: The slanted sides are not parallel because they do not stay the same distance apart.

Answer: The top and bottom sides are parallel.

Let’s compare

• Parallel lines: never meet
• Perpendicular lines: meet to form a right angle

Here is a simple way to remember:

• Parallel = side by side, never touching
• Perpendicular = crossing to make a square corner

Things to watch out for

• If two lines cross but do not make a right angle, they are not perpendicular.
• If two lines look close together but will meet if extended, they are not parallel.
• In shapes, opposite sides are often parallel, and corners with square corners show perpendicular sides.

Practice thinking

1. In a rectangle, which sides are parallel?
2. In a square, what kind of angle is at each corner?
3. If two lines cross and make a square corner, what are they called?
4. If two sides of a shape never meet, what are they called?

Brief Summary

Parallel lines are lines that never meet and stay the same distance apart. Perpendicular lines are lines that meet and form a right angle, which is \(90^\circ\). In polygons, squares and rectangles have both parallel sides and perpendicular sides. Looking for never-meeting lines and square corners helps you identify them correctly.

Attributes of 3D Solids

We see 3D solids all around us. A 3D solid is a shape that is not flat. It has length, width, and height. You can hold it in your hand, like a box, a ball, or a can.

In this lesson, we will learn how to describe 3D solids by their attributes. Attributes are features we can notice and count. For 3D solids, the most important attributes are faces, edges, and vertices.

Here are the three parts to know:

• Face: a flat surface on a solid.
• Edge: the line where two faces meet.
• Vertex: a corner where edges meet. The plural of vertex is vertices.

Some solids also have curved surfaces. A curved surface is smooth and round, not flat. For example, a sphere has no flat faces, but it does have one curved surface.

Let’s learn the main 3D solids.

1. Prism

A prism is a solid with two matching flat ends and flat side faces. A very common prism is a rectangular prism, like a cereal box.

• A rectangular prism has 6 faces, 12 edges, and 8 vertices.
• All of its faces are rectangles.

A cube is a special rectangular prism. All of its faces are squares.

• A cube has 6 faces, 12 edges, and 8 vertices.

2. Pyramid

A pyramid has one base on the bottom and triangle faces that meet at one top point.

A common example is a square pyramid.

• It has 5 faces: 1 square face and 4 triangle faces.
• It has 8 edges.
• It has 5 vertices.

3. Cylinder

A cylinder looks like a can. It has two flat circular faces and one curved surface around the outside.

• It has 2 flat faces.
• It has 1 curved surface.
• It has 0 vertices because it has no corners.

Some lessons also count the circle borders where the flat faces meet the curved surface as edges. In 3rd Grade, it is usually most helpful to remember that a cylinder has no corners and has 2 flat faces and 1 curved surface.

4. Sphere

A sphere looks like a ball.

• It has 0 flat faces.
• It has 1 curved surface.
• It has 0 edges.
• It has 0 vertices.

How to count faces, edges, and vertices

1. Look for all the flat faces.
2. Find where faces meet. These are the edges.
3. Find the corners. These are the vertices.
4. If the solid is round, check for curved surfaces too.

It can help to turn the solid in your mind or use a real object. Sometimes not every face can be seen in a picture, so think about the hidden parts too.

Quick shape facts

• Cube: 6 faces, 12 edges, 8 vertices
• Rectangular prism: 6 faces, 12 edges, 8 vertices
• Square pyramid: 5 faces, 8 edges, 5 vertices
• Cylinder: 2 flat faces, 1 curved surface, 0 vertices
• Sphere: 0 flat faces, 1 curved surface, 0 edges, 0 vertices

Worked Example 1: A cube

Question: How many faces, edges, and vertices does a cube have?

Think: A cube is like a dice or a small box. It has the same number of faces, edges, and vertices as a rectangular prism.

Answer:

• Faces: 6
• Edges: 12
• Vertices: 8

So, a cube has 6 faces, 12 edges, and 8 vertices.

Worked Example 2: A square pyramid

Question: How many faces does a square pyramid have?

Think: It has 1 square base and 4 triangle sides.

We can add them:

$$1 + 4 = 5$$

Answer: A square pyramid has 5 faces.

Now let’s count all its parts:

• Faces: 5
• Edges: 8
• Vertices: 5

Worked Example 3: A cylinder

Question: What attributes does a cylinder have?

Think: A cylinder is like a soup can. It has two flat circular ends and one curved side.

Answer:

• Flat faces: 2
• Curved surfaces: 1
• Vertices: 0

A cylinder has no corners, so it has 0 vertices.

Worked Example 4: Guess the solid

Question: A solid has 0 flat faces, 0 edges, and 0 vertices. What solid is it?

Think: A shape with no flat faces and no corners must be completely round.

Answer: It is a sphere.

Tips for remembering

• Faces are flat.
• Edges are lines where faces meet.
• Vertices are corners.
• A sphere is round all over.
• A cylinder has round ends and no corners.
• A pyramid comes to a point.
• A prism has two matching ends.

Let’s compare solids

A cube and a rectangular prism both have:

• 6 faces
• 12 edges
• 8 vertices

But they are not always the same shape. A cube has all square faces. A rectangular prism has rectangle faces.

A sphere and a cylinder are both round, but they are different too.

• A sphere has no flat faces.
• A cylinder has 2 flat faces.

Summary

3D solids are shapes that take up space. We can describe them by their faces, edges, vertices, and sometimes curved surfaces.

Remember these important facts:

• Cube: 6 faces, 12 edges, 8 vertices
• Rectangular prism: 6 faces, 12 edges, 8 vertices
• Square pyramid: 5 faces, 8 edges, 5 vertices
• Cylinder: 2 flat faces, 1 curved surface, 0 vertices
• Sphere: 0 flat faces, 1 curved surface, 0 edges, 0 vertices

When you look at a 3D solid, ask yourself: How many flat faces do I see? Where do the faces meet? How many corners are there? These questions will help you name and describe the solid correctly.

Composing and Decomposing Shapes means putting shapes together and taking shapes apart.

When we compose shapes, we join smaller shapes to make a bigger shape.

When we decompose shapes, we break a bigger shape into smaller shapes we already know.

This helps us notice how shapes fit together. It also helps us solve problems about shape names, sides, and corners.

Let’s review some basic flat shapes, also called polygons.

• Triangle: 3 sides and 3 corners
• Quadrilateral: 4 sides and 4 corners
• Rectangle: a quadrilateral with 4 right angles
• Square: a rectangle with 4 equal sides
• Pentagon: 5 sides and 5 corners
• Hexagon: 6 sides and 6 corners

We can use these simple shapes like puzzle pieces.

For example, two triangles can be joined to make a bigger shape. A rectangle and a triangle can be joined to make a new shape too.

When shapes are combined, the new shape is called a composite shape.

A composite shape is a shape made from two or more smaller shapes.

How to compose shapes

1. Look at the smaller shapes.
2. Notice their sides and corners.
3. Think about which sides can match up.
4. Put the shapes together to make a larger shape.

How to decompose shapes

1. Look at the whole shape carefully.
2. Try to spot smaller shapes inside it.
3. Draw lines inside the shape to split it.
4. Name the smaller shapes you made.

When you decompose a shape, there may be more than one correct way to split it.

That means two students can draw different lines and both can be correct, as long as the smaller parts are real shapes.

Important idea: The smaller shapes should fit exactly, with no gaps and no overlaps.

If there is a gap, the pieces do not make the whole shape. If pieces overlap, they are covering the same space twice.

Worked Example 1: Compose two triangles

Suppose you have 2 same-size triangles.

If you join them along one side, you can make a quadrilateral.

Sometimes the new shape is a rectangle. Sometimes it is another 4-sided shape.

We can think of it like this:

$$\text{triangle} + \text{triangle} = \text{quadrilateral}$$

The important part is to notice that 2 smaller shapes can make 1 larger shape.

Worked Example 2: Compose a rectangle and a triangle

Imagine a rectangle with a triangle attached to one short side.

This might look like a house shape.

The bottom part is a rectangle, and the roof is a triangle.

Together, they make 1 composite shape.

To describe it, we can say: “This composite shape is made from 1 rectangle and 1 triangle.”

We can write:

$$1\ \text{rectangle} + 1\ \text{triangle} = 1\ \text{composite shape}$$

Worked Example 3: Decompose a hexagon

Now let’s take apart a larger shape.

Suppose you see a hexagon. A hexagon has 6 sides.

You can draw a line inside the hexagon from one corner to another corner.

This can split the hexagon into 2 smaller shapes.

One way is to make 2 trapezoids or other familiar shapes, depending on the hexagon.

Another way, for some hexagons, is to split it into 2 triangles and 1 rectangle.

The key idea is that the whole hexagon can be broken into shapes you know.

Worked Example 4: Decompose an L-shape

An L-shape is a common composite shape.

It looks like two rectangles joined together.

To decompose it, look for a place where you can draw one straight line to split it.

You can often draw a line inside the L-shape to make 2 rectangles.

So we can say: “The L-shape is made of 2 rectangles.”

This is helpful because rectangles are easy to recognize.

Tips for finding smaller shapes inside a larger shape

• Look for corners that make square corners, or right angles.
• Look for long straight sides that can be split into shorter sides.
• Ask yourself, “Do I see a rectangle? A triangle? A square?”
• Try more than one line if the first line does not help.

Let’s think through a few quick questions.

Question 1: If a shape is made from 1 square and 1 triangle, is it a composite shape?

Yes. It is made from more than one smaller shape, so it is a composite shape.

Question 2: Can a rectangle be decomposed?

Yes. You can draw a line from one side to the opposite side and split it into 2 smaller rectangles.

You can also draw a diagonal line from one corner to the opposite corner and split it into 2 triangles.

That shows there can be more than one correct way.

Question 3: If 2 shapes overlap, are they composed correctly?

No. When shapes are composed, they should fit together without overlapping.

What should you say when explaining your thinking?

• “I put these 2 triangles together to make a larger shape.”
• “I drew a line to split the shape into 2 rectangles.”
• “This composite shape is made from a square and a triangle.”
• “There is more than one way to decompose this shape.”

Common mistakes to avoid

• Do not leave gaps between shapes.
• Do not let shapes overlap.
• Do not guess the smaller shapes without checking the sides and corners.
• Do not forget that one shape can sometimes be broken apart in different ways.

Practice thinking

When you see a big shape, ask:

• What smaller shapes do I notice?
• Can I draw a line to split it into familiar shapes?
• Can I put simple shapes together to make this shape?

Composing and decomposing shapes is like working with shape puzzles.

You learn how parts make a whole, and how a whole can be split into parts.

This helps you become stronger at noticing shape patterns and understanding geometry.

Summary

To compose shapes means to join smaller shapes to make a larger shape.

To decompose shapes means to break a larger shape into smaller familiar shapes.

A composite shape is made from two or more shapes, and there may be more than one correct way to decompose a shape.

Always make sure the shapes fit exactly, with no gaps and no overlaps.

Lines of Symmetry are lines that split a shape into two matching halves.

If you could fold the shape on that line and both sides would match up exactly, then that line is a line of symmetry.

This is also called reflective symmetry, because one side looks like a mirror image of the other side.

Let’s learn how to find lines of symmetry in shapes.

What does “matching halves” mean?

The two halves must be the same size and the same shape. When folded, every part on one side must land right on top of the other side.

If even one part does not match, then the line is not a line of symmetry.

How to test for a line of symmetry

1. Look at the shape carefully.
2. Imagine drawing a line through the shape.
3. Pretend to fold the shape on that line.
4. Ask: Do both sides match exactly?

If the answer is yes, the line is a line of symmetry.

If the answer is no, it is not.

Lines of symmetry can go in different directions

• Vertical: up and down
• Horizontal: side to side
• Sometimes a shape can also have a slanted symmetry line

A shape can have:

• 0 lines of symmetry
• 1 line of symmetry
• More than 1 line of symmetry

Important idea: Not every line through the middle of a shape is a line of symmetry. The shape must fold into two perfectly matching parts.

Example 1: A square

A square has 4 equal sides. It has several ways to fold into matching halves.

A square has:

• 1 vertical line of symmetry
• 1 horizontal line of symmetry
• 2 diagonal lines of symmetry

So a square has $$4$$ lines of symmetry.

Worked Example 1

Question: Does a square have a vertical line of symmetry?

Step 1: Imagine a line going straight down the middle.

Step 2: Fold the square on that line.

Step 3: The left half matches the right half exactly.

Answer: Yes. That is a line of symmetry.

Example 2: A rectangle

A rectangle has opposite sides equal. It is not a square unless all 4 sides are equal.

A rectangle has:

• 1 vertical line of symmetry
• 1 horizontal line of symmetry

So a rectangle has $$2$$ lines of symmetry.

The diagonal lines in a rectangle usually do not make matching halves, so they are not lines of symmetry.

Worked Example 2

Question: Does a rectangle have a diagonal line of symmetry?

Step 1: Imagine folding the rectangle from one corner to the opposite corner.

Step 2: Check whether the two halves match exactly.

Step 3: They do not overlap perfectly.

Answer: No. A rectangle does not have diagonal lines of symmetry.

Example 3: An equilateral triangle

An equilateral triangle has 3 equal sides.

It has $$3$$ lines of symmetry.

Each line goes from a corner straight to the middle of the opposite side.

Worked Example 3

Question: How many lines of symmetry does an equilateral triangle have?

Step 1: Test a line from the top corner to the middle of the bottom side.

Step 2: The two sides match.

Step 3: Do the same from each corner.

Step 4: Each one makes matching halves.

Answer: An equilateral triangle has $$3$$ lines of symmetry.

Example 4: A regular pentagon

A regular pentagon has 5 equal sides and 5 equal corners.

A regular pentagon has $$5$$ lines of symmetry.

Each line goes through a corner and the middle of the opposite side.

Worked Example 4

Question: Does a regular pentagon have more than 1 line of symmetry?

Step 1: Test one line through a corner and the middle across from it.

Step 2: The halves match.

Step 3: Try the same idea for the other corners.

Step 4: There are 5 matching fold lines.

Answer: Yes. A regular pentagon has $$5$$ lines of symmetry.

Shapes with 1 line of symmetry

Some shapes have only one line of symmetry.

• A heart often has 1 vertical line of symmetry.
• An isosceles triangle often has 1 vertical line of symmetry.

Shapes with no lines of symmetry

Some shapes cannot be folded into matching halves at all.

• A scalene triangle has 0 lines of symmetry.
• A shape with one side longer or different from the others may have 0 lines of symmetry.

Tips for finding symmetry

• Look for equal sides and equal parts.
• Check whether the left and right sides match.
• Check whether the top and bottom match.
• Do not guess just because a line goes through the center.
• Use the folding test in your mind.

Let’s compare some common shapes

• Square: $$4$$ lines of symmetry
• Rectangle: $$2$$ lines of symmetry
• Equilateral triangle: $$3$$ lines of symmetry
• Regular pentagon: $$5$$ lines of symmetry
• Scalene triangle: $$0$$ lines of symmetry

Real-life symmetry

You can see symmetry in many places around you.

• Butterflies often have a vertical line of symmetry.
• Some leaves have a line of symmetry.
• Windows and tiles can have symmetrical shapes.

Looking for symmetry in real life can help you understand shapes better.

Things to remember

• A line of symmetry makes two identical halves.
• The halves must match exactly when folded.
• A shape can have 0, 1, or many lines of symmetry.
• Test each possible line carefully.

Summary

A line of symmetry is a line that divides a shape into two equal matching halves.

If the shape can fold on the line and both sides overlap perfectly, then the shape has symmetry on that line.

Some shapes, like squares, have many lines of symmetry. Other shapes have only one or none at all.

Partitioning Shapes into Equal Areas

Sometimes in math, we take one whole shape and break it into smaller parts. This is called partitioning a shape.

When we partition a shape into equal areas, each part must be the same size. The parts may look different in some cases, but for this lesson we will mostly use shapes that are split into parts that are both easy to see and equal in size.

This idea helps us understand fractions. If one shape is split into 2 equal parts, each part is one-half, or \(\frac{1}{2}\). If a shape is split into 4 equal parts, each part is one-fourth, or \(\frac{1}{4}\).

Big idea: Equal parts means equal size, not just the same shape or the same number of lines.

For example, if a rectangle is cut into 2 parts and one part is larger than the other, those parts are not equal areas. That means the shape was not partitioned fairly.

Words to know

• Whole – one complete shape
• Partition – to split a shape into parts
• Equal areas – parts that are the same size
• Half – 1 of 2 equal parts, \(\frac{1}{2}\)
• Third – 1 of 3 equal parts, \(\frac{1}{3}\)
• Fourth – 1 of 4 equal parts, \(\frac{1}{4}\)

How to tell if parts are equal

1. Look at the whole shape.
2. Count how many parts it has.
3. Check whether all parts are the same size.
4. If all parts are the same size, the shape is partitioned into equal areas.

You can partition many regular shapes, such as squares, rectangles, and circles. A regular shape has a balanced, even look, which often makes it easier to split into equal parts.

Main teaching point 1: Equal parts must be fair parts

If 2 friends share 1 sandwich equally, each friend should get the same amount. Shapes work the same way in math. Equal areas are like fair shares.

If a square is split down the middle, the 2 pieces are equal. But if the line is closer to one side, one piece will be smaller and one piece will be larger. Those are not equal parts.

Main teaching point 2: The number of equal parts tells the fraction name

• 2 equal parts \(\rightarrow\) halves \(\rightarrow \frac{1}{2}\)
• 3 equal parts \(\rightarrow\) thirds \(\rightarrow \frac{1}{3}\)
• 4 equal parts \(\rightarrow\) fourths \(\rightarrow \frac{1}{4}\)

If a shape has 4 equal parts, one part is not called a half. It is called a fourth because there are 4 equal pieces in the whole.

Main teaching point 3: Equal parts can be shown in different ways

A rectangle can be split into 4 equal parts with vertical lines. It can also be split into 4 equal parts with horizontal lines. Both ways are correct if all parts are equal in size.

So, there is often more than one correct way to partition a shape into equal areas.

Worked Example 1: Partition a rectangle into 2 equal parts

Imagine a rectangle. Draw 1 line right through the middle from top to bottom.

Now the rectangle has 2 parts. Because the line is in the middle, both parts are the same size.

Each part is:

$$\frac{1}{2}$$

That means each part is one-half of the whole rectangle.

Worked Example 2: Is this shape split into equal parts?

A square is split into 2 pieces, but one piece is thin and the other piece is wide.

Ask: Are the 2 parts the same size?

No. One part is bigger.

So this square is not partitioned into equal areas.

Even though there are 2 parts, they are not halves because halves must be equal.

Worked Example 3: Partition a square into 4 equal parts

Start with 1 square. Draw 1 line down the middle. Then draw 1 line across the middle.

Now the square is split into 4 small parts.

Because both lines go through the middle, the 4 parts are equal in size.

Each part is:

$$\frac{1}{4}$$

Each small part is one-fourth of the square.

Worked Example 4: Partition a circle into 4 equal parts

Think of a pizza. If you cut the pizza into 4 same-size slices, each slice is an equal part.

There are 4 equal parts, so each slice is:

$$\frac{1}{4}$$

If one slice is much bigger than the others, then the pizza was not cut into equal areas.

Tips for partitioning shapes

• Try to draw lines through the middle when making halves or fourths.
• Count the parts after you draw the lines.
• Check carefully that the parts are the same size.
• Remember: more parts does not always mean equal parts.

Things to watch out for

• Do not just count pieces. Make sure the pieces are equal.
• Do not call parts halves, thirds, or fourths unless the parts are equal.
• A shape can be divided into parts in different ways and still be correct, as long as the parts are equal in area.

Practice thinking

Ask yourself these questions when you look at a partitioned shape:

• How many parts are there?
• Are all the parts the same size?
• What is the name of one part: half, third, or fourth?

Summary

Partitioning shapes into equal areas means splitting one whole shape into parts that are all the same size. When there are 2 equal parts, each part is \(\frac{1}{2}\). When there are 3 equal parts, each part is \(\frac{1}{3}\). When there are 4 equal parts, each part is \(\frac{1}{4}\).

Always remember: the parts must be equal to use fraction names correctly. If the parts are not the same size, the shape is not partitioned into equal areas.