Exploring Quadrilaterals

This week’s maths lessons we continued with exploring quadrilaterals, building on everything the children discovered last week.

Our focus was not simply remembering shapes, but understanding how the mathematics of quadrilaterals works.

The children explored:

• properties of quadrilaterals
• cube nets
• mathematical vocabulary
• perimeter and area formulas

Most importantly, they discovered why these formulas work, rather than simply memorising them.

This is one of the central goals of living maths and hands on maths—helping children understand the logic behind mathematics rather than simply learning procedures.

Exploring Quadrilaterals: Our Goals for the Week

Our objectives for the week were to:

• Revise the quadrilaterals studied last week
• Discover all eleven cube nets
• Find the perimeter formulas for parallelogram-type quadrilaterals
• Discover the area formulas for parallelograms
• Understand why those formulas work
• Recognise why those formulas do not work for trapeziums or deltoids

By the end of the week, the children would not only know the formulas but understand the geometric reasoning behind them.

Revising Quadrilaterals with a Maths Game

To revise the shapes and vocabulary we had learned, I introduced a simple guessing game.

Each child received a card with the name of a quadrilateral.

I then described one shape using mathematical language, and the girls had to decide:

“Who am I?”

They held up the card that matched the description.

This game also allowed me to introduce a new word into their vocabulary:

Congruent

Congruent describes shapes or lines that are exactly the same size and shape.

The more children hear and use mathematical language naturally, the more confident they become when encountering it later in formal questions.

Guess the Quadrilateral

Next, we reversed the game.

Each child placed a card with a quadrilateral name on their head without looking at it.

They then asked questions to discover which shape they were holding.

For example:

• “Do I have parallel sides?”
• “Do I have right angles?”
• “Are my sides congruent?”

It was a wonderfully engaging way to reinforce vocabulary and properties.

Digital Shape Practice

Each day the girls spent about five minutes on the computer creating the quadrilaterals they had learned.

This quick activity helped them:

• revise shape properties
• reinforce vocabulary
• visualise the shapes again

Short, low-pressure review activities can be incredibly effective for retaining knowledge.

Completing the Eleven Cube Nets

Last week the children discovered six cube nets using magnetic construction pieces.

Since a cube actually has eleven possible nets, I challenged them to find the missing ones.

First, I asked them to try without manipulatives, encouraging them to visualise the shapes.

If they struggled, they were free to use anything helpful.

Naturally, I suggested marshmallows and sticks, which always increases enthusiasm for mathematical investigations.

This challenge required them to think spatially and mentally unfold a cube into two dimensions.

Investigating Perimeter and Area

Next we turned our attention to perimeter and area.

Using large play blocks, the children built two squares of different sizes.

From these shapes we discussed:

• length
• perimeter
• area

In the absence of rulers, the girls chose to measure using:

• blocks for length
• blocks squared for area

This simple decision reinforced the concept of units of measurement.

Rediscovering the Square Formula

Although the children had used these formulas before, I was surprised to discover they had completely forgotten them.

However, they did remember how to calculate the measurements.

Working backwards from their calculations, they rediscovered the formulas for a square:

Perimeter of a square

Perimeter = 4 × side length

Area of a square

Area = side × side

This was a wonderful reminder that rediscovering a formula can be far more powerful than simply memorising it.

Moving to Rectangles

We repeated the same investigation with rectangles.

The children quickly rediscovered:

Perimeter of a rectangle

Perimeter = 2 × (length + width)

Area of a rectangle

Area = length × width

But I asked an important question:

Why does this formula work?

At first they suggested it worked because rectangles have straight sides.

However, triangles also have straight sides.

After some discussion one of them realised the key property:

Right angles.

Squares and rectangles both contain four right angles, which makes calculating area much simpler.

Investigating the Parallelogram

Next we explored the parallelogram and rhombus, which is a special type of parallelogram.

I gave the children:

• cardboard cut-outs of the shapes
• identical shapes printed on squared paper

I asked them how they could calculate perimeter and area.

They suggested:

• measuring with a ruler
• measuring with string
• counting squares

Counting squares works, but it can be very slow.

So I asked an important question:

Could the rectangle formula still work?

They didn’t think it could, mainly because parallelograms don’t have right angles.

Cutting Shapes to Discover the Formula

At this point I gave each child a pair of scissors and told them to figure out the formula.

They looked at me with considerable suspicion.

But soon they began experimenting.

One of the girls folded and cut the parallelogram so that the slanted triangles could be rearranged.

With a little teamwork they discovered something remarkable.

By moving the triangles, they could turn the parallelogram into a rectangle.

Since the new shape was a rectangle, they could apply the familiar formula:

Area = length × width

Understanding Height in a Parallelogram

The children understood that the same formula worked, but they initially assumed the slanted side was the width.

Returning to their cut shapes, they discovered the real width was actually the height—the perpendicular distance between the top and bottom sides.

This led to the standard formula:

Area of a parallelogram

Area = base × height

Although the formula looks slightly different, the mathematics is exactly the same.

I reminded them that letters in formulas are simply representations of numbers.

What matters most is understanding the geometry behind the formula.

Which Quadrilaterals Use These Formulas?

Next, I placed examples of several quadrilaterals on the table.

The girls sorted them into two groups:

1. Shapes where they knew formulas for area and perimeter
2. Shapes where they did not

They quickly noticed something interesting.

All the shapes with formulas were parallelograms.

The others—like the trapezium and deltoid—were not.

This observation naturally sets up our next stage of learning.

After studying triangles, we will return to these shapes and discover the connection between triangle area formulas and non-parallelogram quadrilaterals.

Exploring Quadrilaterals: Testing Their Understanding

To check their understanding, I gave the girls a few questions from a Year 7 (Grade 6) mathematics paper.

T11 flew through the problems quickly.

L10 and C10 worked more slowly but thoughtfully.

They became stuck on one question where the wording was confusing.

In this problem they were given the area and perimeter of a rectangle and asked to determine the length and width.

Once I demonstrated the method, they immediately understood and completed several more examples successfully.

A Huge Change in Confidence

What struck me most during these exercises was the girls’ confidence.

There were no tears.

No frustration.

Even when they struggled, they calmly worked through the problem and listened to explanations.

Their maths anxiety has disappeared.

The hands-on exploration and discovery approach seems to have transformed how they view mathematics.

Reflection Questions for Homeschool Parents

1. Do your children understand why formulas work, or do they simply memorise them?
2. How could cutting or rearranging shapes help children visualise area?
3. Are your children comfortable using mathematical vocabulary in conversation?
4. Do your lessons allow children to rediscover ideas they have previously learned?
5. How might exploring quadrilaterals lead naturally into other geometry topics?

Hands-On Maths Activities to Try

1. Quadrilateral Guessing Game

Describe a quadrilateral using properties. Children hold up the matching shape.

2. Cube Net Challenge

Ask children to find all eleven cube nets using paper or construction toys.

3. Cut-and-Rearrange Geometry

Cut a parallelogram and rearrange it into a rectangle to demonstrate the area formula.

4. Block Measurement

Use building blocks as units to measure perimeter and area.

5. Shape Sorting Investigation

Sort quadrilaterals into groups based on properties such as:

• parallel sides
• right angles
• equal sides

Final Thoughts on Exploring Quadrilaterals

This week’s lesson on exploring quadrilaterals showed once again how powerful hands on maths can be.

Instead of memorising formulas, the children:

• built shapes
• cut shapes
• rearranged shapes
• discovered the mathematics themselves

And in the process, their confidence with geometry grew enormously.

Next week we will move on to square numbers, while continuing to explore area and perimeter in greater depth.

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