Daily doubling fills a pond on day 30. When was it half full?

Day 29. One more doubling changes half into the whole.

64. Multiply three fours together.

What power with base three equals three to the fourth times three cubed?

3^7. Four factors and three factors make seven factors.

What power with base six equals three to the fourth times two to the fourth?

6^4. Each three pairs with a two, making four sixes.

Every dress matches every cap. How many outfits come from four dresses and three caps?

12 outfits. Each of four dresses has three cap choices.

What is seven to the power zero?

1. Dividing equal nonzero powers gives one and subtracts their exponents to zero.

What is 59 times ten squared in standard form?

5.9 × 10^3. The multiplier must be at least one and below ten.

Which is larger: ten to the sixth or ten to the ninth?

Ten to the ninth. The bases match, but nine factors of ten give the larger value.

Teach Power Play in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper, a pencil, two different tops and two different bottoms.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Repeated paper folds double the layers, so the thickness grows quickly.
Clothes make more outfits because each top can pair with different bottoms.
Powers of ten shorten large numbers while keeping their values clear.

What to teach

How to teach it

Repeated multiplication

Unfolded paper beside layer counts and a separate sequence adding one each time.

1. Exponential growth

Positive amounts grow by a fixed multiplier greater than one.

Show: Fold paper in half three times. Record layers: 1, 2, 4, 8. Unfold it. Beside these counts, write a separate sequence labelled 'adding one each time': 1, 2, 3, 4.
Ask: How many layers would a fourth fold make?
Say: Sixteen: each fold doubles all the layers.
Used in: Bacterial growth estimates help food producers plan storage that limits spoilage.

If your child says…“Doubling means adding the same amount each time.”

Ask: Did the layer count increase by the same amount each time?
What’s true: Each increase equals the current amount.

$8 = 2 \times 4$

$8 = 2 \times 2 \times 2$

2. Exponential notation and prime factors

Positive whole-number exponents count repeated factors; prime factorisation groups matching primes.

Show: Write this line. Say, 'Eight is two times four. Split four into two times two. The base repeats; the exponent counts factors.'
Ask: How would you write the next line using a power?
Say: Eight equals two cubed: three twos multiply.
Used in: Prime factorisation finds equal pack sizes without leftovers.

If your child says…“Two cubed means two times three.”

Ask: Are these twos added or multiplied?
What’s true: The exponent counts factors, not additions.

Combining powers and choices

$(2^2)^3 = 2^2 \times 2^2 \times 2^2$

$(2^2)^3 = (2\times2)(2\times2)(2\times2)$

$(2^2)^3 = 2^{2+2+2}$

3. Multiplying powers and powers of powers

Joining factors adds counts; repeating whole groups multiplies counts.

Show: Write this line. Say, 'Multiply three copies of two squared. Each bracket holds two factors.'
Ask: What single power records all the factors of two hidden here?
Say: Six factors of two give two to the sixth. Multiplying powers with matching bases adds their exponents. Repeating a whole power multiplies the inner and outer exponents.
Used in: Repeated image resizing combines scale factors to calculate the final size.

If your child says…“Multiplying powers means multiplying their exponents.”

Ask: How many factors came from each group?
What’s true: When multiplying powers with the same base, keep that base and add the exponents.

$2^2 \times 3^2 = (2 \times 2)(3 \times 3)$

$2^2 \times 3^2 = (2 \times 3)(2 \times 3)$

$2^2 \times 3^2 = 6 \times 6$

4. Powers with the same exponent

Equal exponents allow matching factors to pair.

Show: Write this line. Say, 'Pair each two with a three. Each pair makes six.'
Ask: What single power completes the next line?
Say: The product is six squared because pairing gives two factors of six.
Used in: Area calculations combine squared measurements to compare floor sizes.

If your child says…“Multiply the bases and add the exponents.”

Ask: How many sixes did the pairing produce?
What’s true: Pairing keeps the original number of repetitions.

Two different tops above two different bottoms.

5. How Many Combinations

Multiply choice counts when every pairing is allowed.

Show: Lay tops above bottoms. Keep one top fixed and pair it with each bottom after predicting. Repeat with the other top.
Ask: How many outfits will these clothes make altogether?
Say: Each top makes two outfits, giving four altogether: $2 \times 2 = 2^2$ . Outfits count pairings, not individual garments.
Used in: Combination locks use independent digit choices to provide many possible codes.

If your child says…“Just add the numbers of choices.”

Ask: Does each top get just one bottom, or both?
What’s true: Each top can pair with either bottom, so count both outfits for each top.

Division and powers of ten

$\frac{2^2}{2^2} = \frac{2 \times 2}{2 \times 2}$

$2^{2-2} = \frac{2}{2} \times \frac{2}{2}$

$2^0 = 1 \times 1$

6. The Other Side of Powers

For matching nonzero bases, division subtracts exponents; zero exponents give one.

Show: Write this line. Say, 'Subtracting two from two gives exponent zero.' Cancel matching factors after the prediction.
Ask: What do you predict two to the power zero will equal?
Say: Two to power zero equals one. Each matching pair gives one, and one times one is one.
Used in: Negative powers express small measurements compactly, making precision easier to record.

If your child says…“A zero exponent makes zero.”

Ask: What remains when each fraction equals one?
What’s true: Every nonzero base has zero power equal to one.

$37{,}000 = 3.7 \times 10{,}000$

$10{,}000 = 10^4$

7. Powers of 10 and scientific notation

Standard form uses a multiplier at least one but below ten.

Show: Write this line. Say, 'Ten thousand is ten to the fourth power.'
Ask: How can we finish writing 37,000 compactly using a power of ten?
Say: Thirty-seven thousand is three point seven times ten to the fourth. The multiplier is at least one and below ten. Multiplying by ten thousand restores the original value.
Used in: Scientific notation shortens long distance measurements, making them easier to compare.

If your child says…“Any multiplier works in standard form.”

Ask: Is your multiplier at least one and below ten?
What’s true: An equal value may still need rewriting into standard form.

$10^6 = 1{,}000{,}000 = 1\text{ million}$

$10^9 = 1{,}000{,}000{,}000 = 1\text{ billion}$

$\frac{1{,}000{,}000{,}000}{1{,}000{,}000} = \frac{10^9}{10^6}$

8. Comparing powers and naming large numbers

Powers of ten connect large-number names with their values.

Show: Write this line. Say, 'A million is ten to the sixth power. A billion is ten to the ninth power.'
Ask: How many times as large as a million is a billion?
Say: A billion is one thousand million. Dividing these powers subtracts six from nine, leaving three factors of ten.
Used in: Large-number names help compare population totals written in words.

If your child says…“Only the base matters when comparing powers.”

Ask: Both bases are ten, so what makes their values different?
What’s true: Both the base and exponent affect a power's value.

A real problem

An experiment tested whether a nuclear chain reaction could be created and controlled.

Scientists achieved a self-sustaining nuclear chain reaction on December 2, 1942.

Fermi recognised that neutron production was growing exponentially.

The scientists successfully halted the controlled chain reaction.

University of Chicago Library TIME National Park Service

Upright dominoes arranged in a branching pattern on a tabletop.

Concept Lab · Common misconception

The tempting mistake in Power Play

Open standalone misconception page →

What does an exponent count?

Wrong

2³ = 6

Why it feels right. I multiplied the base by the exponent.

Right

The exponent counts equal factors.

Worked example

2³

2×2×2

The exponent tells you how many copies of 2 to multiply.

Write 2³ as 2 × 2 × 2.
There are three factors and the product is 8.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

Daily doubling fills a pond on day 30. When was it half full?

Day 29. One more doubling changes half into the whole.
What is four cubed?

64. Multiply three fours together.
What power with base three equals three to the fourth times three cubed?

$3^7$ . Four factors and three factors make seven factors.
What power with base six equals three to the fourth times two to the fourth?

$6^4$ . Each three pairs with a two, making four sixes.
Every dress matches every cap. How many outfits come from four dresses and three caps?

12 outfits. Each of four dresses has three cap choices.
What is seven to the power zero?

1. Dividing equal nonzero powers gives one and subtracts their exponents to zero.
What is 59 times ten squared in standard form?

$5.9 \times 10^3$ . The multiplier must be at least one and below ten.
Which is larger: ten to the sixth or ten to the ninth?

Ten to the ninth. The bases match, but nine factors of ten give the larger value.

15 more questions with answers

Teach Power Play in eight steps.

Why it matters