The doubling display

One light is on. Every round, the number of lights doubles. After 5 rounds, is it about 10 lights, or many more?

About 35 minutes
Class 8
Mathematics and money

What you will learn: Write repeated multiplication as a power, see how much faster doubling grows than adding, and multiply powers of the same number.

What should I know first?

If this feels new, try this first. It is fine to go back.

Growing patterns and the nth term Class 7

Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓

Jump to: Check yourself · For grown-ups · For teachers

Safety first

Use paper squares, not real bulbs or wires.

Keep it safe:

Use paper squares, or draw the board. Do not open electric lights or wires.
If you use rice grains, beads or counters, keep them away from babies: nothing goes in the mouth.

You will need

Squared paper and coloured pencilsTo light up squares.
A handful of rice grains or countersOptional, to see doubling in real things.
A calculatorTo check big numbers at the end.

Time: About 35 minutes.

What do you think?

Start with 1 lit square and double 5 times. Tanmay says 10. How many do you think?

Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.

About 10

Let's test it. Colour 1 square, then 2, then 4... Keep a list of each round.

Between 10 and 30

Write each round down: 1, 2, 4, ... What comes after 4? Then check your guess.

More than 30

Check it by writing every round. Then predict round 10 before you work it out.

Not sure

That is fine. Colour 1 square, then double it, and keep going for 5 rounds.

Need a hint?

Doubling means multiplying by 2 each time: 1, 2, 4, ... not adding 2.

The story

For the fair, Tanmay builds a light board out of squares of coloured paper. Round 0: one square lit. Every round he doubles the lit squares. "After 5 rounds that's 10 squares," he says. "Two more each time." Tara shakes her head. "Doubling isn't adding two." Dhruv grabs a pencil. "Let's count it properly before we buy the paper."

The idea

Repeated multiplication of the same number is written as a power: 2 × 2 × 2 × 2 × 2 = 2⁵ ("2 to the power 5").

The small number (the exponent) says how many times the base is multiplied. 2⁵ is not 2 × 5.

Doubling grows very fast: 2¹⁰ = 1,024, while adding 2 ten times to 1 gives only 21.

Multiplying powers of the same base adds the exponents: 2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2⁷.

A bar chart of lit squares after each round: 1 at the start, then 2, 4, 8, 16 and 32 after round 5.

Worked example

Counting the board

How many squares are lit after round 5, and after round 10?

Round 1: 2. Round 2: 4. Round 3: 8. Round 4: 16. Round 5: 32.
That is 2 multiplied 5 times: 2⁵ = 32.
Round 10: 2¹⁰ = 2⁵ × 2⁵ = 32 × 32 = 1,024.

32 squares after round 5, and 1,024 after round 10.

Try it

Light the board.
On squared paper, colour 1 square. Then 2, then 4, doubling each round. Stop when it gets too big to draw.
Keep a table.
Write the round and the number lit. Next to it, write "add 2 each round" for comparison.
Write powers.
Write each doubling number as a power of 2: 8 = 2³, 16 = 2⁴, and so on.
Predict.
Without drawing, predict rounds 6, 7 and 10. Check with a calculator.
Find the law.
Write 2³ × 2⁴ as a long product. How many 2s are there? Try 3² × 3³.
Tell a story.
Write a short story where something doubles: savings, a plant, a forwarded message. When does it get too big?

What you make (paper or offline)

Your doubling chart

A table comparing adding 2 each round with doubling each round, starting from 1: after round 1, 3 and 2; round 3, 7 and 8; round 5, 11 and 32; round 10, 21 and 1,024.

Make your own chart comparing adding and doubling.

Mark the round where doubling overtakes adding.

Write the doubling numbers as powers of 2.

Make it on paper. Keep it with you. Nothing to upload.

What did we find out?

Read this after you have tried it. Look back at your guess and your reason first.

After 5 doublings there are 32 lit squares: 1 × 2 × 2 × 2 × 2 × 2 = 2⁵.

Adding 2 each round gives only 11 after 5 rounds. Doubling quickly races ahead.

Powers of the same number multiply by adding the powers: 2³ × 2⁴ = 2⁷.

Think back

Why does doubling overtake adding, even though it starts slower?

Where in real life does something double again and again?

Investigate further

Fold a sheet of paper in half again and again. How many layers after 7 folds? Can you fold it an 8th time?
Which is bigger, 2¹⁰ or 10²? By how much?

Can you use it here?

If a message is forwarded by every person to 2 new people, how many get it at the 10th step?

Ask someone at home

Ask someone at home: would they rather have ₹1,000 now, or 1 paisa doubled every day for 30 days? Work it out together.

No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.

For grown-upsTap a heading to open it. The safety notes are at the top of the page.

Before you start

Squared paper and coloured pencils. Keep rice grains, beads and counters away from babies; nothing in the mouth.

Questions that help

"Is this adding or multiplying?"
"Predict the next round before you count."
"How many 2s are in this product?"

What's really going on

Children often read 2⁵ as 2 × 5. Writing it out as a long product fixes this.

Doubling (exponential growth) is why savings with interest, populations and viral messages grow so fast.

Adding exponents when multiplying powers comes straight from counting how many times the base appears.

Other ways to do it

No squared paper? Fold a sheet of paper and count the layers.
Use coins: 1, 2, 4, 8 (count them, but do not use real money for big rounds).

For teachersLearning aims, classroom ideas and what to look for.

Learning aims

Write repeated multiplication as a power and evaluate it.
Compare linear (adding) and exponential (doubling) growth.
Explain and use a^m × a^n = a^(m+n) for the same base.

Subject links

NCERT Class 8 Ganita Prakash: powers and exponents (confirm chapter in the current edition).
Topic ladder 4.3 (powers of ten and standard form).
Science: cell division and bacteria (doubling).

These links are general. They have not been checked against a specific board syllabus.

In the classroom

The paper-folding challenge in pairs: predict, then fold and count.
A class chart of "add 2" against "double": find where the lines cross.

What to look for

Reads 2⁵ as five 2s multiplied, not 2 × 5.
Predicts later rounds by multiplying, not adding.
Adds exponents when multiplying powers of the same base.